| L(s) = 1 | − 2-s + 4-s + 1.17·5-s + 1.96·7-s − 8-s − 1.17·10-s + 0.465·11-s − 2.12·13-s − 1.96·14-s + 16-s − 0.590·17-s − 7.81·19-s + 1.17·20-s − 0.465·22-s − 4.36·23-s − 3.62·25-s + 2.12·26-s + 1.96·28-s + 1.44·29-s + 9.39·31-s − 32-s + 0.590·34-s + 2.30·35-s − 3.39·37-s + 7.81·38-s − 1.17·40-s + 12.4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.524·5-s + 0.741·7-s − 0.353·8-s − 0.370·10-s + 0.140·11-s − 0.589·13-s − 0.524·14-s + 0.250·16-s − 0.143·17-s − 1.79·19-s + 0.262·20-s − 0.0992·22-s − 0.911·23-s − 0.724·25-s + 0.416·26-s + 0.370·28-s + 0.267·29-s + 1.68·31-s − 0.176·32-s + 0.101·34-s + 0.388·35-s − 0.558·37-s + 1.26·38-s − 0.185·40-s + 1.93·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 1.17T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 11 | \( 1 - 0.465T + 11T^{2} \) |
| 13 | \( 1 + 2.12T + 13T^{2} \) |
| 17 | \( 1 + 0.590T + 17T^{2} \) |
| 19 | \( 1 + 7.81T + 19T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 - 1.44T + 29T^{2} \) |
| 31 | \( 1 - 9.39T + 31T^{2} \) |
| 37 | \( 1 + 3.39T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 4.71T + 43T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 3.95T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 0.763T + 67T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 + 7.40T + 73T^{2} \) |
| 79 | \( 1 - 0.711T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 + 2.93T + 89T^{2} \) |
| 97 | \( 1 + 6.58T + 97T^{2} \) |
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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
−7.70344101501207038491253845731, −6.80040264840253076055216692617, −6.18740350965008467256919790860, −5.59913014443189067375100107704, −4.53856676851774041417421386224, −4.09581655124006596418291584877, −2.68017164512160849104003322949, −2.16972516218768912214386274239, −1.30557523692591318027236096773, 0,
1.30557523692591318027236096773, 2.16972516218768912214386274239, 2.68017164512160849104003322949, 4.09581655124006596418291584877, 4.53856676851774041417421386224, 5.59913014443189067375100107704, 6.18740350965008467256919790860, 6.80040264840253076055216692617, 7.70344101501207038491253845731
