| L(s) = 1 | + 2-s + 0.825·3-s + 4-s − 2.91·5-s + 0.825·6-s − 1.72·7-s + 8-s − 2.31·9-s − 2.91·10-s + 3.70·11-s + 0.825·12-s − 4.64·13-s − 1.72·14-s − 2.40·15-s + 16-s + 4.45·17-s − 2.31·18-s + 8.38·19-s − 2.91·20-s − 1.42·21-s + 3.70·22-s − 23-s + 0.825·24-s + 3.48·25-s − 4.64·26-s − 4.39·27-s − 1.72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.476·3-s + 0.5·4-s − 1.30·5-s + 0.337·6-s − 0.652·7-s + 0.353·8-s − 0.772·9-s − 0.921·10-s + 1.11·11-s + 0.238·12-s − 1.28·13-s − 0.461·14-s − 0.620·15-s + 0.250·16-s + 1.08·17-s − 0.546·18-s + 1.92·19-s − 0.651·20-s − 0.310·21-s + 0.789·22-s − 0.208·23-s + 0.168·24-s + 0.696·25-s − 0.911·26-s − 0.845·27-s − 0.326·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 0.825T + 3T^{2} \) |
| 5 | \( 1 + 2.91T + 5T^{2} \) |
| 7 | \( 1 + 1.72T + 7T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 19 | \( 1 - 8.38T + 19T^{2} \) |
| 29 | \( 1 - 0.958T + 29T^{2} \) |
| 31 | \( 1 - 0.0262T + 31T^{2} \) |
| 37 | \( 1 - 0.0277T + 37T^{2} \) |
| 41 | \( 1 + 2.57T + 41T^{2} \) |
| 43 | \( 1 - 4.39T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 9.59T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 + 0.166T + 73T^{2} \) |
| 79 | \( 1 + 8.12T + 79T^{2} \) |
| 83 | \( 1 + 4.83T + 83T^{2} \) |
| 89 | \( 1 - 0.889T + 89T^{2} \) |
| 97 | \( 1 - 3.83T + 97T^{2} \) |
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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
−7.64168378419949973527366278559, −7.17822672763395232635568862797, −6.27707897211536773592372673412, −5.50444584734011701254903703096, −4.71532563956417735071196021572, −3.89163329709608239426378288535, −3.16918663811766268809308726639, −2.91888119560383991566059256973, −1.39186878560756025140083623551, 0,
1.39186878560756025140083623551, 2.91888119560383991566059256973, 3.16918663811766268809308726639, 3.89163329709608239426378288535, 4.71532563956417735071196021572, 5.50444584734011701254903703096, 6.27707897211536773592372673412, 7.17822672763395232635568862797, 7.64168378419949973527366278559
