Zeta Function Related Calculations
| zeta(s) | f(n) zeros | g(n) zeros | @0 | @1 | @-1 | @i | (-1,0) | (0,1) | (1,-1) | (1,i) | (i,-1) | (1,2) | (2,3) | (-1,-2) | (-2,-3) | (-1, d) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| z(0) | 0 | 0 | 0 | 1/2 | 1/2 | -1/2 | -1/2 | 1/2 | 0 | -1 | 1 | 3/2 | 5/2 | 3/2 | 5/2 | -1/2 |
| f(n) =
n g(n) = n 2 /2 |
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| z(-1) | 0, -1 | 0, -3/2 | 0 | 5/12 | 1/12 | -1/4 - i/6 | -1/12 | 5/12 | -1/3 | -2/3 - i/6 | 1/3 + i/6 | 23/12 | 53/12 | -5/12 | -23/12 | -1/12 |
| f(n) =
(n 2
+ n)/2 g(n) = n 3 /6 + n 2 /4 |
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| z(-2) | 0, -1, -1/2 | 0, -1 | 0 | 1/3 | 0 | -i/6 | 0 | 1/3 | -1/3 | -1/3 - i/6 | i/6 | 8/3 | 9 | 1/3 | 8/3 | 0 |
| f(n) =
(2n 3
+ 3n 2
+ n)/6 g(n) = n 4 /12 + n 3 /6 + n 2 /12 |
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| z(-3) | 0, -1 | 0 | 0 | 31/120 | -1/120 | 1/8 - i/30 | 1/120 | 31/120 | -4/15 | -2/15 - i/30 | -2/15 + i/30 | 361/120 | 1171/120 | -31/120 | -361/120 | 1/120 |
| f(n) =
(n 4
+ 2n 3
+ n 2
)/4 g(n) = n 5 /20 + n 4 /8 + n 3 /12 |
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| z(-4) | 0, -1, -1/2, (-3±√21)/6 | 0, -1, (-1±√3)/2 | 0 | 1/5 | 0 | 1/15 + i/10 | 0 | 1/5 | -1/5 | -4/15 + i/10 | -1/15 - i/10 | 31/5 | 211/5 | 1/5 | 31/5 | 0 |
| f(n) =
n 5
/5 + n 4
/2 + n 3
/3 - n/30 g(n) = n 6 /30 + n 5 /10 + n 4 /12 - n 2 /60 |
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| z(-5) | 0, -1, (-1±√3)/2 | 0, -3/2, ... | 0 | 41/252 | 1/252 | -1/12 - 5i/63 | -1/252 | 41/252 | -5/21 | -13/42 - 5i/63 | 1/6 + 5i/63 | 1681/504 | 8321/504 | -41/252 | -1681/504 | -1/252 |
| f(n) =
n 6
/6 + n 5
/2 + 5n 4
/12 - n 2
/12 g(n) = n 7 /42 + n 6 /12 + 5n 5 /60 - n 3 /36 |
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| z(-6) | 0, -1, -1/2, ... | 0, -1 | 0 | 1/7 | 0 | -5/42 - i/21 | 0 | 1/7 | -1/7 | -12/42 - i/21 | 5/42 + i/21 | 127/7 | 1399/7 | 1/7 | 127/7 | 0 |
| f(n) =
n 7
/7 + n 6
/2 + n 5
/2 - n 3
/6 + n/42 g(n) = n 8 /56 + n 7 /14 + n 6 /12 - n 4 /24 + n 2 /84 |
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| z(-7) | 0, -1, -1/2, -3/2, ... | 0, -3/2, ... | 0 | 31/240 | -1/240 | 1/16 - 11i/60 | 1/240 | 31/240 | -1/6 | -1/6 - 11i/60 | -1/12 + 11i/60 | 5333/480 | 38161/480 | -31/240 | -5333/480 | 1/240 |
| f(n) =
n 8
/8 + n 7
/2 + 7n 6
/12 - 7n 4
/24 + n 2
/12 g(n) = n 9 /72 + n 8 /16 + n 7 /12 - 7n 5 /120 + n 3 /36 |
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| z(-8) | 0, -1, -1/2, ... | 0, -1 | 0 | 1/9 | 0 | -1/9 - i/18 | 0 | 1/9 | -1/9 | -2/9 - i/18 | 1/9 + i/18 | 511/9 | 8011/9 | 1/9 | 511/9 | 0 |
| f(n) =
n 9
/9 + n 8
/2 + 2n 7
/3 - 7n 5
/15 + 2n 3
/9 - n/30 g(n) = n 10 /90 + n 9 /18 + n 8 /12 - 7n 6 /90 + n 4 /18 - n 2 /60 |
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| z(-9) | 0, -1, (-1±√5)/2, ... | 0, ... | 0 | 61/660 | 1/132 | -1/20 - 7i/165 | -1/132 | 61/660 | -7/55 | -11/44 - 7i/165 | 1/11 + 7i/165 | 20531/1320 | 209821/1320 | -61/660 | -20531/1320 | -1/132 |
| f(n) =
n 10
/10 + n 9
/2 + 3n 8
/4 - 7n 6
/10 + n 4
/2 - 3n 2
/20 g(n) = n 11 /110 + n 10 /20 + 3n 9 /36 - 7n 7 /70 + n 5 /10 - 3n 3 /60 |
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| z(-10) | 0, -1, -1/2, ... | 0, -1 | 0 | 1/11 | 0 | 1/33 - i/22 | 0 | 1/11 | -1/11 | -2/33 - i/22 | -1/33 + i/22 | 2047/11 | 40000+ | 1/11 | 2047/11 | 0 |
| f(n) =
n 11
/11 + n 10
/2 + 5n 9
/6 - n 7
+ n 5
- n 3
/2 + 5n/66 g(n) = n 12 /132 + n 11 /22 + 5n 10 /60 - n 8 /8 + n 6 /6 - n 4 /8 + 5n 2 /132 |
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| z(-11) | 0, -1, ... | 0, ... | 0 | 691/10920 | -691/32760 | ... | 691/32760 | 691/10920 | -691/8190 | ... | ... | ... | ... | ... | ... | 691/32760 |
| f(n) =
n 12
/12 + n 11
/2 + 11n 10
/12 - 11n 8
/8 + 11n 6
/6 - 11n 4
/8 + 5n 2
/12 g(n) = n 13 /156 + n 12 /24 + 11n 11 /132 - 11n 9 /72 + 11n 7 /42 - 11n 5 /40 + 5n 3 /36 |
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