| L(s) = 1 | − 3.26·3-s − 1.74·5-s − 2.44·7-s + 7.67·9-s + 4.91·11-s − 0.951·13-s + 5.70·15-s − 0.287·17-s + 3.32·19-s + 7.99·21-s − 8.64·23-s − 1.95·25-s − 15.2·27-s − 2.58·29-s − 3.10·31-s − 16.0·33-s + 4.26·35-s + 3.76·37-s + 3.10·39-s + 11.7·41-s − 2.03·43-s − 13.3·45-s − 6.06·47-s − 1.01·49-s + 0.937·51-s + 1.73·53-s − 8.58·55-s + ⋯ |
| L(s) = 1 | − 1.88·3-s − 0.780·5-s − 0.924·7-s + 2.55·9-s + 1.48·11-s − 0.263·13-s + 1.47·15-s − 0.0696·17-s + 0.763·19-s + 1.74·21-s − 1.80·23-s − 0.390·25-s − 2.93·27-s − 0.479·29-s − 0.556·31-s − 2.79·33-s + 0.721·35-s + 0.618·37-s + 0.497·39-s + 1.82·41-s − 0.311·43-s − 1.99·45-s − 0.884·47-s − 0.144·49-s + 0.131·51-s + 0.238·53-s − 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 13 | \( 1 + 0.951T + 13T^{2} \) |
| 17 | \( 1 + 0.287T + 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 23 | \( 1 + 8.64T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 + 3.10T + 31T^{2} \) |
| 37 | \( 1 - 3.76T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 + 6.06T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 + 6.43T + 59T^{2} \) |
| 61 | \( 1 - 9.69T + 61T^{2} \) |
| 67 | \( 1 - 5.73T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 - 0.931T + 73T^{2} \) |
| 79 | \( 1 - 9.72T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 3.19T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.38947327875790072200807787318, −6.51842886071631834922769886308, −6.19501484569666212217410275842, −5.59372948887692902939725179517, −4.63988768793539396431348022839, −3.99978450213778917013325194721, −3.52951205615877142545658332049, −1.92949875790122154398842700188, −0.859389848697575378489363217691, 0,
0.859389848697575378489363217691, 1.92949875790122154398842700188, 3.52951205615877142545658332049, 3.99978450213778917013325194721, 4.63988768793539396431348022839, 5.59372948887692902939725179517, 6.19501484569666212217410275842, 6.51842886071631834922769886308, 7.38947327875790072200807787318
