| L(s) = 1 | − 31.2·2-s − 81·3-s + 461.·4-s + 625·5-s + 2.52e3·6-s + 9.73e3·7-s + 1.57e3·8-s + 6.56e3·9-s − 1.95e4·10-s + 2.50e4·11-s − 3.73e4·12-s − 1.29e5·13-s − 3.03e5·14-s − 5.06e4·15-s − 2.85e5·16-s + 5.47e5·17-s − 2.04e5·18-s − 1.30e5·19-s + 2.88e5·20-s − 7.88e5·21-s − 7.80e5·22-s + 1.95e6·23-s − 1.27e5·24-s + 3.90e5·25-s + 4.03e6·26-s − 5.31e5·27-s + 4.49e6·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 0.577·3-s + 0.901·4-s + 0.447·5-s + 0.796·6-s + 1.53·7-s + 0.135·8-s + 0.333·9-s − 0.616·10-s + 0.515·11-s − 0.520·12-s − 1.25·13-s − 2.11·14-s − 0.258·15-s − 1.08·16-s + 1.59·17-s − 0.459·18-s − 0.229·19-s + 0.403·20-s − 0.885·21-s − 0.710·22-s + 1.45·23-s − 0.0784·24-s + 0.200·25-s + 1.73·26-s − 0.192·27-s + 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.471612115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.471612115\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 + 31.2T + 512T^{2} \) |
| 7 | \( 1 - 9.73e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.50e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.29e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.47e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 1.95e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.56e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.33e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.09e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.47e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.78e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.22e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.08e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.97e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.18e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.19e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.40e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.61e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.17e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 7.82e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.91e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.50e8T + 7.60e17T^{2} \) |
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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
−10.12478668201187124185047071161, −9.425672206658819192003268148573, −8.317352065483546705861315197417, −7.63415617102415600601906679879, −6.68183205383416801547955826275, −5.23909902875497721110518336139, −4.56689686122567564974959238230, −2.52606584126199572505725558054, −1.31795610066982129080658122412, −0.824600655623805676111193837215,
0.824600655623805676111193837215, 1.31795610066982129080658122412, 2.52606584126199572505725558054, 4.56689686122567564974959238230, 5.23909902875497721110518336139, 6.68183205383416801547955826275, 7.63415617102415600601906679879, 8.317352065483546705861315197417, 9.425672206658819192003268148573, 10.12478668201187124185047071161
