| L(s) = 1 | − 7.43·2-s − 5.04·3-s + 23.3·4-s − 9.82·5-s + 37.5·6-s − 34.7·7-s + 64.6·8-s − 217.·9-s + 73.0·10-s + 191.·11-s − 117.·12-s + 258.·14-s + 49.5·15-s − 1.22e3·16-s + 64.0·17-s + 1.61e3·18-s + 2.22e3·19-s − 229.·20-s + 175.·21-s − 1.42e3·22-s + 3.17e3·23-s − 326.·24-s − 3.02e3·25-s + 2.32e3·27-s − 808.·28-s − 664.·29-s − 368.·30-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 0.323·3-s + 0.728·4-s − 0.175·5-s + 0.425·6-s − 0.267·7-s + 0.356·8-s − 0.895·9-s + 0.231·10-s + 0.476·11-s − 0.235·12-s + 0.351·14-s + 0.0569·15-s − 1.19·16-s + 0.0537·17-s + 1.17·18-s + 1.41·19-s − 0.128·20-s + 0.0866·21-s − 0.626·22-s + 1.25·23-s − 0.115·24-s − 0.969·25-s + 0.613·27-s − 0.194·28-s − 0.146·29-s − 0.0748·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 7.43T + 32T^{2} \) |
| 3 | \( 1 + 5.04T + 243T^{2} \) |
| 5 | \( 1 + 9.82T + 3.12e3T^{2} \) |
| 7 | \( 1 + 34.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 191.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 64.0T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.22e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 664.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.20e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.06e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.75e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.65e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5T + 8.58e9T^{2} \) |
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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
−11.31494845147883430310896251876, −10.16553756609634907847838965901, −9.333031253583771473659582363976, −8.438147472828328729054057617235, −7.44659793843632251975874623534, −6.29741445560361094213930410897, −4.87967709981767071375163174961, −3.08849677018877832340199348832, −1.25223988569556360587474205291, 0,
1.25223988569556360587474205291, 3.08849677018877832340199348832, 4.87967709981767071375163174961, 6.29741445560361094213930410897, 7.44659793843632251975874623534, 8.438147472828328729054057617235, 9.333031253583771473659582363976, 10.16553756609634907847838965901, 11.31494845147883430310896251876
