| L(s) = 1 | − 41.3·2-s − 243·3-s − 336.·4-s − 4.78e3·5-s + 1.00e4·6-s + 1.68e4·7-s + 9.86e4·8-s + 5.90e4·9-s + 1.97e5·10-s + 8.84e5·11-s + 8.17e4·12-s − 3.71e5·13-s − 6.95e5·14-s + 1.16e6·15-s − 3.39e6·16-s + 6.33e6·17-s − 2.44e6·18-s − 7.31e6·19-s + 1.60e6·20-s − 4.08e6·21-s − 3.65e7·22-s − 1.47e7·23-s − 2.39e7·24-s − 2.59e7·25-s + 1.53e7·26-s − 1.43e7·27-s − 5.65e6·28-s + ⋯ |
| L(s) = 1 | − 0.914·2-s − 0.577·3-s − 0.164·4-s − 0.684·5-s + 0.527·6-s + 0.377·7-s + 1.06·8-s + 0.333·9-s + 0.625·10-s + 1.65·11-s + 0.0948·12-s − 0.277·13-s − 0.345·14-s + 0.395·15-s − 0.808·16-s + 1.08·17-s − 0.304·18-s − 0.677·19-s + 0.112·20-s − 0.218·21-s − 1.51·22-s − 0.477·23-s − 0.614·24-s − 0.531·25-s + 0.253·26-s − 0.192·27-s − 0.0620·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 7 | \( 1 - 1.68e4T \) |
| 13 | \( 1 + 3.71e5T \) |
| good | 2 | \( 1 + 41.3T + 2.04e3T^{2} \) |
| 5 | \( 1 + 4.78e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 8.84e5T + 2.85e11T^{2} \) |
| 17 | \( 1 - 6.33e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 7.31e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.47e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.00e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.95e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.53e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.20e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.41e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.91e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 6.36e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.83e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.25e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.37e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.32e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.27e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.47e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.01e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.43e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.31e11T + 7.15e21T^{2} \) |
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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
−9.486457288778566381334481380169, −8.613457855505893849927366442396, −7.71614522202428098677356698492, −6.88765758079980729669063283043, −5.63509064243823561283159872239, −4.39714724062248738838275688524, −3.74714718361636092875840064679, −1.78462523510510460654984329700, −0.949065967146797810703919260956, 0,
0.949065967146797810703919260956, 1.78462523510510460654984329700, 3.74714718361636092875840064679, 4.39714724062248738838275688524, 5.63509064243823561283159872239, 6.88765758079980729669063283043, 7.71614522202428098677356698492, 8.613457855505893849927366442396, 9.486457288778566381334481380169
