| L(s) = 1 | − 4.09e3·2-s + 1.47e6·3-s + 1.67e7·4-s − 6.04e9·6-s − 3.57e9·7-s − 6.87e10·8-s + 1.32e12·9-s + 1.08e12·11-s + 2.47e13·12-s − 2.13e13·13-s + 1.46e13·14-s + 2.81e14·16-s + 3.79e15·17-s − 5.43e15·18-s + 6.90e15·19-s − 5.27e15·21-s − 4.44e15·22-s + 1.56e17·23-s − 1.01e17·24-s + 8.73e16·26-s + 7.08e17·27-s − 6.00e16·28-s + 6.08e17·29-s − 3.42e18·31-s − 1.15e18·32-s + 1.60e18·33-s − 1.55e19·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.60·3-s + 0.5·4-s − 1.13·6-s − 0.0977·7-s − 0.353·8-s + 1.56·9-s + 0.104·11-s + 0.801·12-s − 0.253·13-s + 0.0691·14-s + 0.250·16-s + 1.57·17-s − 1.10·18-s + 0.715·19-s − 0.156·21-s − 0.0737·22-s + 1.48·23-s − 0.566·24-s + 0.179·26-s + 0.907·27-s − 0.0488·28-s + 0.319·29-s − 0.782·31-s − 0.176·32-s + 0.167·33-s − 1.11·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(3.604844729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.604844729\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.47e6T + 8.47e11T^{2} \) |
| 7 | \( 1 + 3.57e9T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.08e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 2.13e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 3.79e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 6.90e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.56e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 6.08e17T + 3.63e36T^{2} \) |
| 31 | \( 1 + 3.42e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 7.61e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.17e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 4.70e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 6.53e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.54e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.39e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 4.30e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 3.04e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 4.08e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 1.14e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 6.98e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.18e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 2.79e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 1.06e25T + 4.66e49T^{2} \) |
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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
−10.41733667112266166366015277756, −9.500764326324518817873433082286, −8.746213761039097463534423497432, −7.75517708081166202534425351081, −7.01155614330686515424598522013, −5.27978378460535468542022702426, −3.57633042962747265535331697554, −2.95093618486670167139879099364, −1.81097538074598414604652979047, −0.848819215835978688464876235594,
0.848819215835978688464876235594, 1.81097538074598414604652979047, 2.95093618486670167139879099364, 3.57633042962747265535331697554, 5.27978378460535468542022702426, 7.01155614330686515424598522013, 7.75517708081166202534425351081, 8.746213761039097463534423497432, 9.500764326324518817873433082286, 10.41733667112266166366015277756
