| L(s) = 1 | − 1.83·2-s − 11.9·3-s − 28.6·4-s + 21.9·6-s − 112.·7-s + 111.·8-s − 100.·9-s + 162.·11-s + 341.·12-s + 169·13-s + 206.·14-s + 711.·16-s − 379.·17-s + 185.·18-s − 284.·19-s + 1.33e3·21-s − 299.·22-s + 4.18e3·23-s − 1.32e3·24-s − 310.·26-s + 4.09e3·27-s + 3.21e3·28-s + 6.27e3·29-s − 7.55e3·31-s − 4.87e3·32-s − 1.94e3·33-s + 697.·34-s + ⋯ |
| L(s) = 1 | − 0.324·2-s − 0.765·3-s − 0.894·4-s + 0.248·6-s − 0.865·7-s + 0.615·8-s − 0.414·9-s + 0.405·11-s + 0.684·12-s + 0.277·13-s + 0.281·14-s + 0.694·16-s − 0.318·17-s + 0.134·18-s − 0.180·19-s + 0.661·21-s − 0.131·22-s + 1.65·23-s − 0.470·24-s − 0.0901·26-s + 1.08·27-s + 0.773·28-s + 1.38·29-s − 1.41·31-s − 0.841·32-s − 0.310·33-s + 0.103·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 1.83T + 32T^{2} \) |
| 3 | \( 1 + 11.9T + 243T^{2} \) |
| 7 | \( 1 + 112.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 162.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 379.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 284.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.06e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.16e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.15e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.77e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.65e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.58e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.49e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 561.T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.12e4T + 8.58e9T^{2} \) |
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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
−10.41391486160668984635370733001, −9.148454679607861526615007893044, −8.824415704862137295758706667795, −7.36559270049813849118169697315, −6.30772447135876289305919091828, −5.38112871652369070745587107739, −4.30210116310734126445653076622, −3.03878643102663109794731415565, −1.02225915352418629172568426778, 0,
1.02225915352418629172568426778, 3.03878643102663109794731415565, 4.30210116310734126445653076622, 5.38112871652369070745587107739, 6.30772447135876289305919091828, 7.36559270049813849118169697315, 8.824415704862137295758706667795, 9.148454679607861526615007893044, 10.41391486160668984635370733001
