| L(s) = 1 | + 4.09e3·2-s − 1.37e6·3-s + 1.67e7·4-s + 8.78e8·5-s − 5.64e9·6-s + 3.00e10·7-s + 6.87e10·8-s + 1.05e12·9-s + 3.60e12·10-s + 5.73e12·11-s − 2.31e13·12-s − 1.07e13·13-s + 1.23e14·14-s − 1.21e15·15-s + 2.81e14·16-s + 2.97e15·17-s + 4.30e15·18-s − 5.42e15·19-s + 1.47e16·20-s − 4.13e16·21-s + 2.34e16·22-s − 1.04e17·23-s − 9.46e16·24-s + 4.74e17·25-s − 4.39e16·26-s − 2.81e17·27-s + 5.03e17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.49·3-s + 0.5·4-s + 1.61·5-s − 1.05·6-s + 0.820·7-s + 0.353·8-s + 1.24·9-s + 1.13·10-s + 0.551·11-s − 0.748·12-s − 0.127·13-s + 0.580·14-s − 2.41·15-s + 0.250·16-s + 1.23·17-s + 0.877·18-s − 0.562·19-s + 0.805·20-s − 1.22·21-s + 0.389·22-s − 0.994·23-s − 0.529·24-s + 1.59·25-s − 0.0903·26-s − 0.361·27-s + 0.410·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(2.212283994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212283994\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| good | 3 | \( 1 + 1.37e6T + 8.47e11T^{2} \) |
| 5 | \( 1 - 8.78e8T + 2.98e17T^{2} \) |
| 7 | \( 1 - 3.00e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 5.73e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.07e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 2.97e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 5.42e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.04e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 3.09e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.26e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 4.51e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 7.56e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.39e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 4.67e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 1.24e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.32e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.36e20T + 4.29e44T^{2} \) |
| 67 | \( 1 + 5.36e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 2.27e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 2.78e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 6.36e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.19e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 3.22e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 3.58e24T + 4.66e49T^{2} \) |
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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
−22.15134100365496802842364456197, −21.19208389753462498356108644001, −17.86971452131425100101498252999, −16.80071504643086272845410878330, −14.14472070512823609426422755833, −12.15149674897595662697810873644, −10.39165734775282512659645327459, −6.25928827265899712557255199877, −5.10664108876688748126544343749, −1.54121893824147299728369205257,
1.54121893824147299728369205257, 5.10664108876688748126544343749, 6.25928827265899712557255199877, 10.39165734775282512659645327459, 12.15149674897595662697810873644, 14.14472070512823609426422755833, 16.80071504643086272845410878330, 17.86971452131425100101498252999, 21.19208389753462498356108644001, 22.15134100365496802842364456197
