| L(s) = 1 | + 659.·2-s + 6.56e3·3-s + 3.03e5·4-s − 1.08e6·5-s + 4.32e6·6-s + 1.59e6·7-s + 1.13e8·8-s + 4.30e7·9-s − 7.13e8·10-s − 4.47e8·11-s + 1.99e9·12-s − 2.48e9·13-s + 1.05e9·14-s − 7.10e9·15-s + 3.50e10·16-s − 2.48e10·17-s + 2.83e10·18-s + 8.23e10·19-s − 3.28e11·20-s + 1.04e10·21-s − 2.94e11·22-s + 6.43e11·23-s + 7.44e11·24-s + 4.10e11·25-s − 1.63e12·26-s + 2.82e11·27-s + 4.84e11·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 0.577·3-s + 2.31·4-s − 1.24·5-s + 1.05·6-s + 0.104·7-s + 2.39·8-s + 0.333·9-s − 2.25·10-s − 0.629·11-s + 1.33·12-s − 0.844·13-s + 0.190·14-s − 0.715·15-s + 2.04·16-s − 0.864·17-s + 0.606·18-s + 1.11·19-s − 2.87·20-s + 0.0605·21-s − 1.14·22-s + 1.71·23-s + 1.38·24-s + 0.537·25-s − 1.53·26-s + 0.192·27-s + 0.242·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(4.032098581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.032098581\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| good | 2 | \( 1 - 659.T + 1.31e5T^{2} \) |
| 5 | \( 1 + 1.08e6T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.59e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 4.47e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.48e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 2.48e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 8.23e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 6.43e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 9.82e11T + 7.25e24T^{2} \) |
| 31 | \( 1 - 3.28e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.63e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 3.33e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 9.83e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.62e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 1.40e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 9.80e13T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.37e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 1.85e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 6.17e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.30e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.27e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.42e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.77e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.09e17T + 5.95e33T^{2} \) |
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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
−22.41705788410736049892210111130, −20.77973404631493090688209368255, −19.54454648704164377391957063196, −15.84042662563218817911405150332, −14.78912087820957412594154867832, −13.05073225793903038042554351315, −11.47466914490790494379001657764, −7.39551238701347746065126219882, −4.66470067397013088175077662639, −2.97393553239677131736204792795,
2.97393553239677131736204792795, 4.66470067397013088175077662639, 7.39551238701347746065126219882, 11.47466914490790494379001657764, 13.05073225793903038042554351315, 14.78912087820957412594154867832, 15.84042662563218817911405150332, 19.54454648704164377391957063196, 20.77973404631493090688209368255, 22.41705788410736049892210111130
