| L(s) = 1 | + 1.32·2-s − 0.242·4-s + 1.07·5-s − 7-s − 2.97·8-s + 1.43·10-s + 4.78·11-s − 3.48·13-s − 1.32·14-s − 3.45·16-s + 2.77·17-s − 2.57·19-s − 0.261·20-s + 6.34·22-s + 0.614·23-s − 3.83·25-s − 4.62·26-s + 0.242·28-s + 7.90·29-s − 3.61·31-s + 1.36·32-s + 3.67·34-s − 1.07·35-s − 6.08·37-s − 3.41·38-s − 3.20·40-s − 4.86·41-s + ⋯ |
| L(s) = 1 | + 0.937·2-s − 0.121·4-s + 0.482·5-s − 0.377·7-s − 1.05·8-s + 0.452·10-s + 1.44·11-s − 0.967·13-s − 0.354·14-s − 0.864·16-s + 0.672·17-s − 0.591·19-s − 0.0585·20-s + 1.35·22-s + 0.128·23-s − 0.766·25-s − 0.906·26-s + 0.0458·28-s + 1.46·29-s − 0.649·31-s + 0.241·32-s + 0.630·34-s − 0.182·35-s − 0.999·37-s − 0.554·38-s − 0.507·40-s − 0.760·41-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 | \( 1 - 1.32T + 2T^{2} \) |
| 5 | \( 1 - 1.07T + 5T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 - 2.77T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 - 0.614T + 23T^{2} \) |
| 29 | \( 1 - 7.90T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 + 8.64T + 47T^{2} \) |
| 53 | \( 1 - 8.77T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 + 0.747T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 - 0.421T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 7.71T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−7.23731304457424530855076965385, −6.51021805192309263396767947260, −6.13784486377830978531130183536, −5.25660816672739552016086737644, −4.72905648338342041310658056581, −3.88872480398530795216470632005, −3.34307760934448155560612241770, −2.43656491703378826447533057125, −1.40658512589450135471306099959, 0,
1.40658512589450135471306099959, 2.43656491703378826447533057125, 3.34307760934448155560612241770, 3.88872480398530795216470632005, 4.72905648338342041310658056581, 5.25660816672739552016086737644, 6.13784486377830978531130183536, 6.51021805192309263396767947260, 7.23731304457424530855076965385
