| L(s) = 1 | − 7.78·2-s − 81·3-s − 451.·4-s + 625·5-s + 630.·6-s + 1.83e3·7-s + 7.50e3·8-s + 6.56e3·9-s − 4.86e3·10-s + 4.43e4·11-s + 3.65e4·12-s + 1.36e5·13-s − 1.43e4·14-s − 5.06e4·15-s + 1.72e5·16-s − 2.53e5·17-s − 5.10e4·18-s + 8.54e4·19-s − 2.82e5·20-s − 1.49e5·21-s − 3.45e5·22-s − 9.79e5·23-s − 6.07e5·24-s + 3.90e5·25-s − 1.06e6·26-s − 5.31e5·27-s − 8.30e5·28-s + ⋯ |
| L(s) = 1 | − 0.344·2-s − 0.577·3-s − 0.881·4-s + 0.447·5-s + 0.198·6-s + 0.289·7-s + 0.647·8-s + 0.333·9-s − 0.153·10-s + 0.914·11-s + 0.508·12-s + 1.32·13-s − 0.0996·14-s − 0.258·15-s + 0.658·16-s − 0.736·17-s − 0.114·18-s + 0.150·19-s − 0.394·20-s − 0.167·21-s − 0.314·22-s − 0.729·23-s − 0.373·24-s + 0.200·25-s − 0.456·26-s − 0.192·27-s − 0.255·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.114088960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114088960\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| good | 2 | \( 1 + 7.78T + 512T^{2} \) |
| 7 | \( 1 - 1.83e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.43e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.36e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.53e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.54e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 9.79e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.94e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.56e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.44e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.27e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.16e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.70e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.35e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.48e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.10e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.43e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.19e9T + 7.60e17T^{2} \) |
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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
−17.50879103791912782747065835903, −16.14276913180374231901880793526, −14.25474033065510341558678295690, −13.11268265924733439960737401695, −11.36294620416704591200972006337, −9.834405788870403866626550613586, −8.439094873599952929310718602775, −6.20036582165188861355970623800, −4.35095565650486616548205522556, −1.08218705022902846707049609956,
1.08218705022902846707049609956, 4.35095565650486616548205522556, 6.20036582165188861355970623800, 8.439094873599952929310718602775, 9.834405788870403866626550613586, 11.36294620416704591200972006337, 13.11268265924733439960737401695, 14.25474033065510341558678295690, 16.14276913180374231901880793526, 17.50879103791912782747065835903
