| L(s) = 1 | + 6.63·2-s − 0.104·3-s + 12.0·4-s − 25·5-s − 0.691·6-s − 82.7·7-s − 132.·8-s − 242.·9-s − 165.·10-s − 121·11-s − 1.25·12-s − 1.15e3·13-s − 549.·14-s + 2.60·15-s − 1.26e3·16-s − 1.60e3·17-s − 1.61e3·18-s + 361·19-s − 301.·20-s + 8.61·21-s − 803.·22-s − 2.23e3·23-s + 13.7·24-s + 625·25-s − 7.69e3·26-s + 50.6·27-s − 997.·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s − 0.00668·3-s + 0.376·4-s − 0.447·5-s − 0.00783·6-s − 0.638·7-s − 0.731·8-s − 0.999·9-s − 0.524·10-s − 0.301·11-s − 0.00251·12-s − 1.90·13-s − 0.748·14-s + 0.00298·15-s − 1.23·16-s − 1.34·17-s − 1.17·18-s + 0.229·19-s − 0.168·20-s + 0.00426·21-s − 0.353·22-s − 0.882·23-s + 0.00488·24-s + 0.200·25-s − 2.23·26-s + 0.0133·27-s − 0.240·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3449479591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3449479591\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 6.63T + 32T^{2} \) |
| 3 | \( 1 + 0.104T + 243T^{2} \) |
| 7 | \( 1 + 82.7T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.15e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.60e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.22e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.37e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.37e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.79e5T + 8.58e9T^{2} \) |
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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
−9.230739603636758439367630097488, −8.334322128699583086805538348372, −7.32994570565699790010048758286, −6.43109785052415766849944647393, −5.63893391408010866627757203039, −4.75464102727670972753677569857, −4.09056116186884280168977135290, −2.87506832416563365607552427298, −2.46373776816923026781149312651, −0.19269376995781381808525477807,
0.19269376995781381808525477807, 2.46373776816923026781149312651, 2.87506832416563365607552427298, 4.09056116186884280168977135290, 4.75464102727670972753677569857, 5.63893391408010866627757203039, 6.43109785052415766849944647393, 7.32994570565699790010048758286, 8.334322128699583086805538348372, 9.230739603636758439367630097488
