| L(s) = 1 | + 4.01e3·2-s − 3.88e5·3-s + 7.74e6·4-s − 1.56e9·6-s − 3.81e9·7-s − 2.59e9·8-s + 5.67e10·9-s + 2.52e11·11-s − 3.00e12·12-s + 3.59e12·13-s − 1.53e13·14-s − 7.53e13·16-s − 2.34e14·17-s + 2.27e14·18-s − 6.23e14·19-s + 1.48e15·21-s + 1.01e15·22-s + 3.58e15·23-s + 1.00e15·24-s + 1.44e16·26-s + 1.45e16·27-s − 2.95e16·28-s − 2.05e16·29-s + 1.36e17·31-s − 2.80e17·32-s − 9.79e16·33-s − 9.40e17·34-s + ⋯ |
| L(s) = 1 | + 1.38·2-s − 1.26·3-s + 0.922·4-s − 1.75·6-s − 0.728·7-s − 0.106·8-s + 0.602·9-s + 0.266·11-s − 1.16·12-s + 0.555·13-s − 1.01·14-s − 1.07·16-s − 1.65·17-s + 0.835·18-s − 1.22·19-s + 0.922·21-s + 0.369·22-s + 0.785·23-s + 0.135·24-s + 0.770·26-s + 0.502·27-s − 0.672·28-s − 0.313·29-s + 0.963·31-s − 1.37·32-s − 0.337·33-s − 2.29·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(1.717536787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717536787\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 4.01e3T + 8.38e6T^{2} \) |
| 3 | \( 1 + 3.88e5T + 9.41e10T^{2} \) |
| 7 | \( 1 + 3.81e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 2.52e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 3.59e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.34e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 6.23e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 3.58e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 2.05e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.36e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.23e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 1.40e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 2.18e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 8.67e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 7.63e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.01e18T + 5.36e40T^{2} \) |
| 61 | \( 1 - 2.87e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.47e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 7.64e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 3.49e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 1.02e22T + 4.42e43T^{2} \) |
| 83 | \( 1 - 7.71e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 4.58e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.13e23T + 4.96e45T^{2} \) |
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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
−12.74582234142462004227932436701, −11.61165657927483258112748298807, −10.76210504882136473605865528192, −8.971196529638716008857195859792, −6.58602291235129800034266331427, −6.19248536121156329932400571951, −4.89612160098352117364581473701, −3.95997718386367537560736170234, −2.48577793966964915075129149878, −0.54623411542559162509864494023,
0.54623411542559162509864494023, 2.48577793966964915075129149878, 3.95997718386367537560736170234, 4.89612160098352117364581473701, 6.19248536121156329932400571951, 6.58602291235129800034266331427, 8.971196529638716008857195859792, 10.76210504882136473605865528192, 11.61165657927483258112748298807, 12.74582234142462004227932436701
