| L(s) = 1 | − 3.38·2-s + 3.43·4-s + 15.4·8-s − 79.6·16-s − 87.3·17-s + 102.·19-s − 121.·23-s + 337.·31-s + 146.·32-s + 295.·34-s − 345.·38-s + 411.·46-s − 545.·47-s − 343·49-s + 706.·53-s + 943.·61-s − 1.14e3·62-s + 143.·64-s − 300.·68-s + 351.·76-s − 1.33e3·79-s − 1.34e3·83-s − 418.·92-s + 1.84e3·94-s + 1.16e3·98-s − 2.39e3·106-s + 17.8·107-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.429·4-s + 0.681·8-s − 1.24·16-s − 1.24·17-s + 1.23·19-s − 1.10·23-s + 1.95·31-s + 0.806·32-s + 1.48·34-s − 1.47·38-s + 1.32·46-s − 1.69·47-s − 49-s + 1.83·53-s + 1.98·61-s − 2.33·62-s + 0.280·64-s − 0.535·68-s + 0.529·76-s − 1.90·79-s − 1.78·83-s − 0.474·92-s + 2.02·94-s + 1.19·98-s − 2.19·106-s + 0.0161·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.38T + 8T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 337.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 545.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 706.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 943.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
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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
−9.829783246399789311537842009313, −8.717782648537857779501607139854, −8.189579413409213655875564314034, −7.23792175591640841738977788167, −6.38366066166691029180531573994, −5.05941728130878856054463035392, −4.06160621953882815794829083132, −2.52878013313946510713583833451, −1.26508012409001347136225239196, 0,
1.26508012409001347136225239196, 2.52878013313946510713583833451, 4.06160621953882815794829083132, 5.05941728130878856054463035392, 6.38366066166691029180531573994, 7.23792175591640841738977788167, 8.189579413409213655875564314034, 8.717782648537857779501607139854, 9.829783246399789311537842009313
