| L(s) = 1 | − 4·2-s − 24.3·3-s + 16·4-s + 97.4·6-s − 49·7-s − 64·8-s + 351.·9-s + 56.6·11-s − 389.·12-s + 899.·13-s + 196·14-s + 256·16-s − 1.49e3·17-s − 1.40e3·18-s + 1.83e3·19-s + 1.19e3·21-s − 226.·22-s − 3.97e3·23-s + 1.55e3·24-s − 3.59e3·26-s − 2.63e3·27-s − 784·28-s + 3.68e3·29-s − 5.88e3·31-s − 1.02e3·32-s − 1.38e3·33-s + 5.97e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.56·3-s + 0.5·4-s + 1.10·6-s − 0.377·7-s − 0.353·8-s + 1.44·9-s + 0.141·11-s − 0.781·12-s + 1.47·13-s + 0.267·14-s + 0.250·16-s − 1.25·17-s − 1.02·18-s + 1.16·19-s + 0.590·21-s − 0.0998·22-s − 1.56·23-s + 0.552·24-s − 1.04·26-s − 0.694·27-s − 0.188·28-s + 0.813·29-s − 1.10·31-s − 0.176·32-s − 0.220·33-s + 0.886·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5566576994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5566576994\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 24.3T + 243T^{2} \) |
| 11 | \( 1 - 56.6T + 1.61e5T^{2} \) |
| 13 | \( 1 - 899.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.49e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.83e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.97e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.68e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 314.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.76e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.89e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.84e5T + 8.58e9T^{2} \) |
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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
−10.79266118493637161188423077219, −9.915951924020141364473122297763, −8.930166186178364454966418916810, −7.77647579583258439510551298869, −6.47784731745359928411127556094, −6.18498326237848452658387825728, −4.95289671784463437865460225674, −3.58165689919066951361501330394, −1.67558940162115346720618090069, −0.49999906246854165055659466961,
0.49999906246854165055659466961, 1.67558940162115346720618090069, 3.58165689919066951361501330394, 4.95289671784463437865460225674, 6.18498326237848452658387825728, 6.47784731745359928411127556094, 7.77647579583258439510551298869, 8.930166186178364454966418916810, 9.915951924020141364473122297763, 10.79266118493637161188423077219
