| L(s) = 1 | + 8.19·2-s − 28.0·3-s + 35.1·4-s − 229.·6-s + 14.0·7-s + 26.1·8-s + 544.·9-s + 567.·11-s − 987.·12-s + 298.·13-s + 115.·14-s − 911.·16-s − 2.20e3·17-s + 4.45e3·18-s + 1.26e3·19-s − 395.·21-s + 4.65e3·22-s + 1.61e3·23-s − 734.·24-s + 2.44e3·26-s − 8.44e3·27-s + 495.·28-s − 5.63e3·29-s + 961·31-s − 8.31e3·32-s − 1.59e4·33-s − 1.80e4·34-s + ⋯ |
| L(s) = 1 | + 1.44·2-s − 1.79·3-s + 1.09·4-s − 2.60·6-s + 0.108·7-s + 0.144·8-s + 2.23·9-s + 1.41·11-s − 1.97·12-s + 0.490·13-s + 0.157·14-s − 0.890·16-s − 1.84·17-s + 3.24·18-s + 0.805·19-s − 0.195·21-s + 2.04·22-s + 0.637·23-s − 0.260·24-s + 0.710·26-s − 2.22·27-s + 0.119·28-s − 1.24·29-s + 0.179·31-s − 1.43·32-s − 2.54·33-s − 2.67·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.531235948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.531235948\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - 961T \) |
| good | 2 | \( 1 - 8.19T + 32T^{2} \) |
| 3 | \( 1 + 28.0T + 243T^{2} \) |
| 7 | \( 1 - 14.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 567.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 298.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.61e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.63e3T + 2.05e7T^{2} \) |
| 37 | \( 1 - 8.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.02e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.44e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 526.T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.71e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.96e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.82e4T + 8.58e9T^{2} \) |
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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
−9.766068422156676054218618945339, −8.842545453079400340491290768685, −7.07551672485057393587255063035, −6.56720475524146491846287512082, −5.92256315749316598709159281787, −5.05567752685850744291881985122, −4.37989549151574613843915709797, −3.59651021032607876611054198930, −1.87293172270271701206357009811, −0.65141096277056900940246027649,
0.65141096277056900940246027649, 1.87293172270271701206357009811, 3.59651021032607876611054198930, 4.37989549151574613843915709797, 5.05567752685850744291881985122, 5.92256315749316598709159281787, 6.56720475524146491846287512082, 7.07551672485057393587255063035, 8.842545453079400340491290768685, 9.766068422156676054218618945339
