| L(s) = 1 | + 3.43e3·2-s − 1.62e5·3-s + 3.42e6·4-s − 1.45e8·5-s − 5.59e8·6-s − 1.70e10·8-s − 6.76e10·9-s − 5.01e11·10-s + 1.15e12·11-s − 5.57e11·12-s + 8.83e10·13-s + 2.37e13·15-s − 8.73e13·16-s + 2.16e14·17-s − 2.32e14·18-s + 8.54e14·19-s − 4.99e14·20-s + 3.98e15·22-s + 8.54e14·23-s + 2.77e15·24-s + 9.36e15·25-s + 3.03e14·26-s + 2.63e16·27-s + 1.01e14·29-s + 8.16e16·30-s − 4.00e16·31-s − 1.57e17·32-s + ⋯ |
| L(s) = 1 | + 1.18·2-s − 0.530·3-s + 0.408·4-s − 1.33·5-s − 0.630·6-s − 0.702·8-s − 0.718·9-s − 1.58·10-s + 1.22·11-s − 0.216·12-s + 0.0136·13-s + 0.709·15-s − 1.24·16-s + 1.52·17-s − 0.852·18-s + 1.68·19-s − 0.545·20-s + 1.45·22-s + 0.187·23-s + 0.372·24-s + 0.785·25-s + 0.0162·26-s + 0.912·27-s + 0.00154·29-s + 0.841·30-s − 0.283·31-s − 0.771·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 3.43e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 1.62e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.45e8T + 1.19e16T^{2} \) |
| 11 | \( 1 - 1.15e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 8.83e10T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.16e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 8.54e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 8.54e14T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.01e14T + 4.31e33T^{2} \) |
| 31 | \( 1 + 4.00e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.48e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 2.26e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 1.87e17T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.98e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 9.67e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 2.17e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 4.33e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 3.42e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 1.52e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.68e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 6.89e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.03e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 3.54e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 6.64e22T + 4.96e45T^{2} \) |
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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
−11.32688526810229811347514474342, −9.488933845265946869881596162963, −8.185523817513365960664283807960, −6.93469756625369294391051800658, −5.72188900495852908523909502345, −4.87703970865276295612624857678, −3.63455495109075774858117643202, −3.20510070307321427148758352976, −1.09047601714530645230925872911, 0,
1.09047601714530645230925872911, 3.20510070307321427148758352976, 3.63455495109075774858117643202, 4.87703970865276295612624857678, 5.72188900495852908523909502345, 6.93469756625369294391051800658, 8.185523817513365960664283807960, 9.488933845265946869881596162963, 11.32688526810229811347514474342
