| L(s) = 1 | + 4.17e3·2-s − 1.12e6·3-s − 1.61e7·4-s + 2.44e8·5-s − 4.70e9·6-s − 2.36e10·7-s − 2.07e11·8-s + 4.26e11·9-s + 1.01e12·10-s + 4.01e12·11-s + 1.82e13·12-s + 1.43e14·13-s − 9.86e13·14-s − 2.75e14·15-s − 3.23e14·16-s − 2.93e14·17-s + 1.77e15·18-s + 1.13e16·19-s − 3.94e15·20-s + 2.66e16·21-s + 1.67e16·22-s + 6.23e16·23-s + 2.34e17·24-s + 5.96e16·25-s + 6.00e17·26-s + 4.75e17·27-s + 3.81e17·28-s + ⋯ |
| L(s) = 1 | + 0.720·2-s − 1.22·3-s − 0.481·4-s + 0.447·5-s − 0.883·6-s − 0.645·7-s − 1.06·8-s + 0.502·9-s + 0.322·10-s + 0.385·11-s + 0.589·12-s + 1.71·13-s − 0.465·14-s − 0.548·15-s − 0.287·16-s − 0.122·17-s + 0.362·18-s + 1.17·19-s − 0.215·20-s + 0.791·21-s + 0.277·22-s + 0.593·23-s + 1.30·24-s + 0.199·25-s + 1.23·26-s + 0.609·27-s + 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(1.410749317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410749317\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 2.44e8T \) |
| good | 2 | \( 1 - 4.17e3T + 3.35e7T^{2} \) |
| 3 | \( 1 + 1.12e6T + 8.47e11T^{2} \) |
| 7 | \( 1 + 2.36e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 4.01e12T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.43e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 2.93e14T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.13e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 6.23e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 3.04e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 1.32e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 6.21e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.60e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 4.09e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 3.28e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 2.96e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 2.05e20T + 1.86e44T^{2} \) |
| 61 | \( 1 - 1.08e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 1.15e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 2.39e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.69e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 1.73e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 5.78e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.93e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 4.55e24T + 4.66e49T^{2} \) |
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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
−17.42476247379819253592276125574, −15.93443370907134569372542645110, −13.89302124684572008036159186282, −12.65656814500521349968300076791, −11.13948856564613783896630073132, −9.229984475166647966721695584263, −6.31674493816719350806421433900, −5.37082563808613739320078952296, −3.58761407238452785473078968339, −0.805196777266114034413544588637,
0.805196777266114034413544588637, 3.58761407238452785473078968339, 5.37082563808613739320078952296, 6.31674493816719350806421433900, 9.229984475166647966721695584263, 11.13948856564613783896630073132, 12.65656814500521349968300076791, 13.89302124684572008036159186282, 15.93443370907134569372542645110, 17.42476247379819253592276125574
