| L(s) = 1 | − 14.8·2-s − 5.24·3-s + 93.5·4-s − 195.·5-s + 78.0·6-s + 343·7-s + 513.·8-s − 2.15e3·9-s + 2.91e3·10-s − 3.18e3·11-s − 490.·12-s + 2.19e3·13-s − 5.10e3·14-s + 1.02e3·15-s − 1.96e4·16-s + 6.85e3·17-s + 3.21e4·18-s − 4.25e4·19-s − 1.82e4·20-s − 1.79e3·21-s + 4.73e4·22-s − 6.56e4·23-s − 2.69e3·24-s − 3.98e4·25-s − 3.26e4·26-s + 2.28e4·27-s + 3.20e4·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 0.112·3-s + 0.730·4-s − 0.700·5-s + 0.147·6-s + 0.377·7-s + 0.354·8-s − 0.987·9-s + 0.921·10-s − 0.721·11-s − 0.0819·12-s + 0.277·13-s − 0.497·14-s + 0.0785·15-s − 1.19·16-s + 0.338·17-s + 1.29·18-s − 1.42·19-s − 0.511·20-s − 0.0424·21-s + 0.948·22-s − 1.12·23-s − 0.0397·24-s − 0.509·25-s − 0.364·26-s + 0.222·27-s + 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3957124887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3957124887\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 + 14.8T + 128T^{2} \) |
| 3 | \( 1 + 5.24T + 2.18e3T^{2} \) |
| 5 | \( 1 + 195.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.18e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 6.85e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.25e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.56e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.47e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.14e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.12e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.01e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.15e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.37e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.50e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.58e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.86e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.16e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.29e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.18e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.25e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.14e5T + 8.07e13T^{2} \) |
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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
−12.38433485240147715876602939794, −11.16495171547365674912320389288, −10.58290629515415408791440911313, −9.173073592181881468474972548787, −8.218761960684568642379118920083, −7.57093612098084437506713269942, −5.86224582454841038259526513759, −4.17837614689931972531957025067, −2.23568639822199071509877170261, −0.47044971842319153896974196468,
0.47044971842319153896974196468, 2.23568639822199071509877170261, 4.17837614689931972531957025067, 5.86224582454841038259526513759, 7.57093612098084437506713269942, 8.218761960684568642379118920083, 9.173073592181881468474972548787, 10.58290629515415408791440911313, 11.16495171547365674912320389288, 12.38433485240147715876602939794
