| L(s) = 1 | − 17.9·2-s + 192.·4-s − 84.6·5-s + 1.67e3·7-s − 1.16e3·8-s + 1.51e3·10-s + 1.33e3·11-s + 1.46e4·13-s − 3.00e4·14-s − 3.88e3·16-s + 2.88e4·17-s − 2.06e4·19-s − 1.63e4·20-s − 2.38e4·22-s − 2.44e4·23-s − 7.09e4·25-s − 2.61e5·26-s + 3.23e5·28-s − 4.48e4·29-s − 7.97e4·31-s + 2.18e5·32-s − 5.15e5·34-s − 1.42e5·35-s + 1.14e5·37-s + 3.70e5·38-s + 9.83e4·40-s + 1.24e5·41-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.50·4-s − 0.302·5-s + 1.85·7-s − 0.801·8-s + 0.479·10-s + 0.301·11-s + 1.84·13-s − 2.92·14-s − 0.236·16-s + 1.42·17-s − 0.691·19-s − 0.456·20-s − 0.477·22-s − 0.418·23-s − 0.908·25-s − 2.92·26-s + 2.78·28-s − 0.341·29-s − 0.481·31-s + 1.17·32-s − 2.25·34-s − 0.560·35-s + 0.372·37-s + 1.09·38-s + 0.242·40-s + 0.282·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.215639570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215639570\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 + 17.9T + 128T^{2} \) |
| 5 | \( 1 + 84.6T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.67e3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 1.46e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.88e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.06e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.44e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.48e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.97e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.14e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.24e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.82e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.17e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.08e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.64e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.70e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.01e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.89e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.30e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.43e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.84e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.81e6T + 8.07e13T^{2} \) |
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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
−11.88983662777939150943137509682, −11.15452542881603587786812436171, −10.40067299657388150436438834360, −8.945023570830742222260715094473, −8.198115411998613734861340847600, −7.51607301205623378594817513679, −5.82953041806940574223323266024, −4.05976281573639098541481190373, −1.80978238853595178445888082932, −0.961508728400052722850077834357,
0.961508728400052722850077834357, 1.80978238853595178445888082932, 4.05976281573639098541481190373, 5.82953041806940574223323266024, 7.51607301205623378594817513679, 8.198115411998613734861340847600, 8.945023570830742222260715094473, 10.40067299657388150436438834360, 11.15452542881603587786812436171, 11.88983662777939150943137509682
