L(s) = 1 | − 2·4-s + 3.56·5-s − 7-s − 3.12·11-s − 3.56·13-s + 4·16-s − 6.56·17-s + 4·19-s − 7.12·20-s + 4.68·23-s + 7.68·25-s + 2·28-s − 4.12·29-s + 4.68·31-s − 3.56·35-s − 1.87·37-s + 10.8·41-s − 4.24·43-s + 6.24·44-s + 4.87·47-s + 49-s + 7.12·52-s + 11·53-s − 11.1·55-s + 4.43·59-s − 9.56·61-s − 8·64-s + ⋯ |
L(s) = 1 | − 4-s + 1.59·5-s − 0.377·7-s − 0.941·11-s − 0.987·13-s + 16-s − 1.59·17-s + 0.917·19-s − 1.59·20-s + 0.976·23-s + 1.53·25-s + 0.377·28-s − 0.765·29-s + 0.841·31-s − 0.602·35-s − 0.308·37-s + 1.68·41-s − 0.647·43-s + 0.941·44-s + 0.711·47-s + 0.142·49-s + 0.987·52-s + 1.51·53-s − 1.49·55-s + 0.577·59-s − 1.22·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 - 11T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 + 9.56T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 + 0.315T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 + 7.80T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.38429401110820330868565780220, −6.82313560103423260722374516800, −5.87276776489624215003881744529, −5.38780045626242777932726458878, −4.85778870256030746362785242912, −4.06561941990325323286361794382, −2.78875097811252789394803703666, −2.44418355168446324462678173059, −1.22037487584328205368848134808, 0,
1.22037487584328205368848134808, 2.44418355168446324462678173059, 2.78875097811252789394803703666, 4.06561941990325323286361794382, 4.85778870256030746362785242912, 5.38780045626242777932726458878, 5.87276776489624215003881744529, 6.82313560103423260722374516800, 7.38429401110820330868565780220