| L(s) = 1 | − 1.18·2-s − 0.601·4-s − 4.39·5-s + 7-s + 3.07·8-s + 5.20·10-s + 1.66·11-s + 1.04·13-s − 1.18·14-s − 2.43·16-s − 2.31·17-s − 1.96·19-s + 2.64·20-s − 1.96·22-s − 2.53·23-s + 14.3·25-s − 1.23·26-s − 0.601·28-s − 3.94·29-s + 2.35·31-s − 3.27·32-s + 2.73·34-s − 4.39·35-s − 3.81·37-s + 2.32·38-s − 13.5·40-s − 5.25·41-s + ⋯ |
| L(s) = 1 | − 0.836·2-s − 0.300·4-s − 1.96·5-s + 0.377·7-s + 1.08·8-s + 1.64·10-s + 0.501·11-s + 0.288·13-s − 0.316·14-s − 0.609·16-s − 0.561·17-s − 0.450·19-s + 0.591·20-s − 0.419·22-s − 0.528·23-s + 2.86·25-s − 0.241·26-s − 0.113·28-s − 0.733·29-s + 0.423·31-s − 0.578·32-s + 0.469·34-s − 0.743·35-s − 0.627·37-s + 0.376·38-s − 2.13·40-s − 0.820·41-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 | \( 1 + 1.18T + 2T^{2} \) |
| 5 | \( 1 + 4.39T + 5T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 17 | \( 1 + 2.31T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 23 | \( 1 + 2.53T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 2.35T + 31T^{2} \) |
| 37 | \( 1 + 3.81T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + 6.13T + 53T^{2} \) |
| 59 | \( 1 - 1.48T + 59T^{2} \) |
| 61 | \( 1 - 0.959T + 61T^{2} \) |
| 67 | \( 1 - 4.40T + 67T^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 - 0.363T + 79T^{2} \) |
| 83 | \( 1 - 9.34T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.68303241012477671868704694017, −7.12928856872857446910468872526, −6.39546746120966935875445354170, −5.15853195436850417559875363555, −4.46892378294435880594886862300, −3.98473293230687688843046363730, −3.33991375941013502710182245482, −1.96830099579191750490943233214, −0.872644942956393490708974784664, 0,
0.872644942956393490708974784664, 1.96830099579191750490943233214, 3.33991375941013502710182245482, 3.98473293230687688843046363730, 4.46892378294435880594886862300, 5.15853195436850417559875363555, 6.39546746120966935875445354170, 7.12928856872857446910468872526, 7.68303241012477671868704694017
