| L(s) = 1 | − 659.·2-s + 3.03e5·4-s + 1.08e6·5-s + 1.59e6·7-s − 1.13e8·8-s − 7.13e8·10-s + 4.47e8·11-s − 2.48e9·13-s − 1.05e9·14-s + 3.50e10·16-s + 2.48e10·17-s + 8.23e10·19-s + 3.28e11·20-s − 2.94e11·22-s − 6.43e11·23-s + 4.10e11·25-s + 1.63e12·26-s + 4.84e11·28-s + 9.82e11·29-s + 3.28e12·31-s − 8.23e12·32-s − 1.63e13·34-s + 1.73e12·35-s + 2.63e13·37-s − 5.42e13·38-s − 1.22e14·40-s + 3.33e13·41-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.31·4-s + 1.24·5-s + 0.104·7-s − 2.39·8-s − 2.25·10-s + 0.629·11-s − 0.844·13-s − 0.190·14-s + 2.04·16-s + 0.864·17-s + 1.11·19-s + 2.87·20-s − 1.14·22-s − 1.71·23-s + 0.537·25-s + 1.53·26-s + 0.242·28-s + 0.364·29-s + 0.692·31-s − 1.32·32-s − 1.57·34-s + 0.129·35-s + 1.23·37-s − 2.02·38-s − 2.96·40-s + 0.651·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(1.035918424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035918424\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 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 | \( 1 - 1.09e17T + 5.95e33T^{2} \) |
| 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
−17.34414913779403351298239903316, −16.19446765847558636690099321161, −14.25263409098795832740525477833, −11.90020173081925787600603400079, −10.12014993243829902696537758119, −9.407295597891158730038533639508, −7.69529648333991374460729900145, −6.04832504035247431722767009581, −2.33131230755794192488502596700, −1.00561375945946341752624147608,
1.00561375945946341752624147608, 2.33131230755794192488502596700, 6.04832504035247431722767009581, 7.69529648333991374460729900145, 9.407295597891158730038533639508, 10.12014993243829902696537758119, 11.90020173081925787600603400079, 14.25263409098795832740525477833, 16.19446765847558636690099321161, 17.34414913779403351298239903316
