| L(s) = 1 | − 10.3·2-s + 9·3-s + 75.5·4-s + 33.2·5-s − 93.3·6-s − 56.9·7-s − 451.·8-s + 81·9-s − 344.·10-s + 432.·11-s + 680.·12-s + 178.·13-s + 590.·14-s + 299.·15-s + 2.26e3·16-s + 1.65e3·17-s − 840.·18-s − 168.·19-s + 2.51e3·20-s − 512.·21-s − 4.48e3·22-s + 143.·23-s − 4.06e3·24-s − 2.01e3·25-s − 1.84e3·26-s + 729·27-s − 4.30e3·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.577·3-s + 2.36·4-s + 0.594·5-s − 1.05·6-s − 0.439·7-s − 2.49·8-s + 0.333·9-s − 1.09·10-s + 1.07·11-s + 1.36·12-s + 0.292·13-s + 0.805·14-s + 0.343·15-s + 2.21·16-s + 1.39·17-s − 0.611·18-s − 0.107·19-s + 1.40·20-s − 0.253·21-s − 1.97·22-s + 0.0564·23-s − 1.44·24-s − 0.646·25-s − 0.535·26-s + 0.192·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.229535091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.229535091\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 + 10.3T + 32T^{2} \) |
| 5 | \( 1 - 33.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 56.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 432.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 178.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.65e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 168.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 143.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 275.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.27e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.65e3T + 4.18e8T^{2} \) |
| 61 | \( 1 - 2.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.24e4T + 8.58e9T^{2} \) |
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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
−11.48119377947168429934401953828, −10.28996013320686687164480522584, −9.627575623768760848129252127585, −8.956022896720903408890873363016, −7.954504056934920708469417046334, −6.94088846164375724928511798380, −5.93285946839259268775284678545, −3.47441571646836258728613302216, −2.03692070308396394438693293033, −0.936215857751623622924571040945,
0.936215857751623622924571040945, 2.03692070308396394438693293033, 3.47441571646836258728613302216, 5.93285946839259268775284678545, 6.94088846164375724928511798380, 7.954504056934920708469417046334, 8.956022896720903408890873363016, 9.627575623768760848129252127585, 10.28996013320686687164480522584, 11.48119377947168429934401953828
