| L(s) = 1 | − 1.26e5·2-s − 8.63e7·3-s + 7.38e9·4-s + 1.52e11·5-s + 1.09e13·6-s + 7.78e13·7-s + 1.52e14·8-s + 1.89e15·9-s − 1.92e16·10-s + 1.61e17·11-s − 6.37e17·12-s − 2.28e18·13-s − 9.84e18·14-s − 1.31e19·15-s − 8.26e19·16-s + 2.46e20·17-s − 2.39e20·18-s − 1.53e21·19-s + 1.12e21·20-s − 6.72e21·21-s − 2.04e22·22-s − 2.16e22·23-s − 1.31e22·24-s + 2.32e22·25-s + 2.89e23·26-s + 3.16e23·27-s + 5.75e23·28-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 1.15·3-s + 0.859·4-s + 0.447·5-s + 1.57·6-s + 0.885·7-s + 0.190·8-s + 0.340·9-s − 0.609·10-s + 1.06·11-s − 0.995·12-s − 0.953·13-s − 1.20·14-s − 0.517·15-s − 1.12·16-s + 1.22·17-s − 0.464·18-s − 1.22·19-s + 0.384·20-s − 1.02·21-s − 1.44·22-s − 0.735·23-s − 0.221·24-s + 0.200·25-s + 1.30·26-s + 0.763·27-s + 0.761·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(0.6489982189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6489982189\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 1.52e11T \) |
| good | 2 | \( 1 + 1.26e5T + 8.58e9T^{2} \) |
| 3 | \( 1 + 8.63e7T + 5.55e15T^{2} \) |
| 7 | \( 1 - 7.78e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 1.61e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.28e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.46e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.53e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.16e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 2.07e23T + 1.81e48T^{2} \) |
| 31 | \( 1 - 1.75e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 8.12e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 1.92e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 8.01e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 7.53e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 1.75e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 2.97e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 3.70e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 8.43e29T + 1.82e60T^{2} \) |
| 71 | \( 1 - 5.92e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 7.49e30T + 3.08e61T^{2} \) |
| 79 | \( 1 - 6.07e30T + 4.18e62T^{2} \) |
| 83 | \( 1 - 3.47e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.80e32T + 2.13e64T^{2} \) |
| 97 | \( 1 - 1.09e33T + 3.65e65T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.76199098162569760365475918922, −14.44994961692951512737681442002, −12.05305686940475542548586740387, −10.85245258198327515084305817331, −9.607200677611740859295500562729, −8.002717772307227546656617874337, −6.36637790558535352432706740019, −4.75978964464573230976129542578, −1.81880437593070529066471579048, −0.66056961295763754977032736231,
0.66056961295763754977032736231, 1.81880437593070529066471579048, 4.75978964464573230976129542578, 6.36637790558535352432706740019, 8.002717772307227546656617874337, 9.607200677611740859295500562729, 10.85245258198327515084305817331, 12.05305686940475542548586740387, 14.44994961692951512737681442002, 16.76199098162569760365475918922
