| L(s) = 1 | − 11.3·2-s − 618.·3-s − 8.06e3·4-s + 4.11e4·5-s + 6.98e3·6-s − 1.34e5·7-s + 1.83e5·8-s − 1.21e6·9-s − 4.64e5·10-s − 8.15e6·11-s + 4.98e6·12-s − 4.55e6·13-s + 1.51e6·14-s − 2.54e7·15-s + 6.39e7·16-s + 1.53e8·17-s + 1.37e7·18-s − 3.29e7·19-s − 3.31e8·20-s + 8.29e7·21-s + 9.22e7·22-s + 1.48e8·23-s − 1.13e8·24-s + 4.69e8·25-s + 5.14e7·26-s + 1.73e9·27-s + 1.08e9·28-s + ⋯ |
| L(s) = 1 | − 0.124·2-s − 0.489·3-s − 0.984·4-s + 1.17·5-s + 0.0611·6-s − 0.431·7-s + 0.247·8-s − 0.760·9-s − 0.146·10-s − 1.38·11-s + 0.481·12-s − 0.261·13-s + 0.0538·14-s − 0.576·15-s + 0.953·16-s + 1.54·17-s + 0.0949·18-s − 0.160·19-s − 1.15·20-s + 0.211·21-s + 0.173·22-s + 0.208·23-s − 0.121·24-s + 0.385·25-s + 0.0326·26-s + 0.861·27-s + 0.424·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.085436748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085436748\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 1.48e8T \) |
| good | 2 | \( 1 + 11.3T + 8.19e3T^{2} \) |
| 3 | \( 1 + 618.T + 1.59e6T^{2} \) |
| 5 | \( 1 - 4.11e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 1.34e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 8.15e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 4.55e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.53e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.29e7T + 4.20e16T^{2} \) |
| 29 | \( 1 - 5.09e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 1.65e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.18e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.99e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.08e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 7.92e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.16e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 3.82e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 1.29e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.12e12T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.26e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 5.84e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 5.91e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.01e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 3.55e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 2.47e12T + 6.73e25T^{2} \) |
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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
−14.45742025807735368978895472961, −13.49942897859516242738616924944, −12.42575718236977563410894972920, −10.45637972380379450645564837979, −9.615611798113143722017164796998, −8.105134028699367800115921979985, −5.94645320035820361645340905815, −5.06944810477653660823172355516, −2.81961629635370257100452443495, −0.71344810860913431844948385790,
0.71344810860913431844948385790, 2.81961629635370257100452443495, 5.06944810477653660823172355516, 5.94645320035820361645340905815, 8.105134028699367800115921979985, 9.615611798113143722017164796998, 10.45637972380379450645564837979, 12.42575718236977563410894972920, 13.49942897859516242738616924944, 14.45742025807735368978895472961
