| L(s) = 1 | − 18.7·2-s + 59.7·3-s + 222.·4-s − 1.11e3·6-s − 438.·7-s − 1.76e3·8-s + 1.38e3·9-s + 5.75e3·11-s + 1.32e4·12-s + 3.53e3·13-s + 8.20e3·14-s + 4.59e3·16-s + 2.39e4·17-s − 2.58e4·18-s + 1.65e4·19-s − 2.61e4·21-s − 1.07e5·22-s + 6.56e4·23-s − 1.05e5·24-s − 6.60e4·26-s − 4.80e4·27-s − 9.74e4·28-s + 1.34e5·29-s + 1.29e5·31-s + 1.40e5·32-s + 3.44e5·33-s − 4.49e5·34-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.27·3-s + 1.73·4-s − 2.11·6-s − 0.482·7-s − 1.21·8-s + 0.631·9-s + 1.30·11-s + 2.21·12-s + 0.445·13-s + 0.798·14-s + 0.280·16-s + 1.18·17-s − 1.04·18-s + 0.554·19-s − 0.616·21-s − 2.15·22-s + 1.12·23-s − 1.55·24-s − 0.737·26-s − 0.470·27-s − 0.838·28-s + 1.02·29-s + 0.777·31-s + 0.755·32-s + 1.66·33-s − 1.95·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.203265469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203265469\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 18.7T + 128T^{2} \) |
| 3 | \( 1 - 59.7T + 2.18e3T^{2} \) |
| 7 | \( 1 + 438.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.75e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.53e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.39e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.65e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.56e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.61e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.88e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.43e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.66e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.54e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.52e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.56e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.99e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.12e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.95e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.21e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.20e6T + 8.07e13T^{2} \) |
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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
−16.28227778878984763170578910710, −14.96493434470865914425534579469, −13.71254450604755438732266872508, −11.77457545858674323883667981207, −10.05681379015378481096806964839, −9.124198961950055201304904540142, −8.216885360980605847364266559377, −6.81653541048283097726178791896, −3.18976327602921800453378511147, −1.27417314287542001573427560230,
1.27417314287542001573427560230, 3.18976327602921800453378511147, 6.81653541048283097726178791896, 8.216885360980605847364266559377, 9.124198961950055201304904540142, 10.05681379015378481096806964839, 11.77457545858674323883667981207, 13.71254450604755438732266872508, 14.96493434470865914425534579469, 16.28227778878984763170578910710
