| L(s) = 1 | − 2·2-s + 8.17·3-s + 4·4-s + 18.4·5-s − 16.3·6-s + 7·7-s − 8·8-s + 39.8·9-s − 36.8·10-s + 31.5·11-s + 32.6·12-s − 14·14-s + 150.·15-s + 16·16-s + 106.·17-s − 79.6·18-s − 107.·19-s + 73.6·20-s + 57.2·21-s − 63.1·22-s − 86.5·23-s − 65.3·24-s + 213.·25-s + 104.·27-s + 28·28-s − 174.·29-s − 300.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s + 1.64·5-s − 1.11·6-s + 0.377·7-s − 0.353·8-s + 1.47·9-s − 1.16·10-s + 0.865·11-s + 0.786·12-s − 0.267·14-s + 2.58·15-s + 0.250·16-s + 1.52·17-s − 1.04·18-s − 1.29·19-s + 0.822·20-s + 0.594·21-s − 0.611·22-s − 0.784·23-s − 0.556·24-s + 1.70·25-s + 0.747·27-s + 0.188·28-s − 1.12·29-s − 1.83·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.120093554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.120093554\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 8.17T + 27T^{2} \) |
| 5 | \( 1 - 18.4T + 125T^{2} \) |
| 11 | \( 1 - 31.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 86.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 39.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 188.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 472.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 681.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 389.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 222.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 47.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 792.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 935.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 113.T + 9.12e5T^{2} \) |
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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
−8.874696658409414432941803154207, −7.929542643777246275724423253561, −7.48629480407434162297927127195, −6.24428903033572481286122170517, −5.86694974521486058751398161307, −4.46566341447165566498125419368, −3.45718919838811334936020246087, −2.45848884925892731226714644366, −1.89851674707037128414034574532, −1.13301715860990150614052334150,
1.13301715860990150614052334150, 1.89851674707037128414034574532, 2.45848884925892731226714644366, 3.45718919838811334936020246087, 4.46566341447165566498125419368, 5.86694974521486058751398161307, 6.24428903033572481286122170517, 7.48629480407434162297927127195, 7.929542643777246275724423253561, 8.874696658409414432941803154207
