| L(s) = 1 | − 4·2-s − 2.52·3-s + 16·4-s + 10.1·6-s + 49·7-s − 64·8-s − 236.·9-s − 267.·11-s − 40.4·12-s − 896.·13-s − 196·14-s + 256·16-s − 61.1·17-s + 946.·18-s + 1.62e3·19-s − 123.·21-s + 1.07e3·22-s + 4.28e3·23-s + 161.·24-s + 3.58e3·26-s + 1.21e3·27-s + 784·28-s − 7.42e3·29-s − 8.93e3·31-s − 1.02e3·32-s + 677.·33-s + 244.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.162·3-s + 0.5·4-s + 0.114·6-s + 0.377·7-s − 0.353·8-s − 0.973·9-s − 0.667·11-s − 0.0811·12-s − 1.47·13-s − 0.267·14-s + 0.250·16-s − 0.0513·17-s + 0.688·18-s + 1.03·19-s − 0.0613·21-s + 0.471·22-s + 1.68·23-s + 0.0573·24-s + 1.03·26-s + 0.320·27-s + 0.188·28-s − 1.63·29-s − 1.67·31-s − 0.176·32-s + 0.108·33-s + 0.0363·34-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
| \(L(3)\) |
\(\approx\) |
\(0.8726239107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8726239107\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.52T + 243T^{2} \) |
| 11 | \( 1 + 267.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 896.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 61.1T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.62e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.42e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.93e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 640.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.87e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.97e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.97e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.99e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.21e5T + 8.58e9T^{2} \) |
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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
−10.81252226507216044774908645000, −9.545562058268836328979601409514, −8.979557837356359118952650746872, −7.68899614471290168864108519925, −7.24901748483705289097621979862, −5.69370182602562433079523799994, −5.00386198706595367086984655004, −3.18470245009348320806537757475, −2.14174020402757731799802499259, −0.54630923308262662775265383008,
0.54630923308262662775265383008, 2.14174020402757731799802499259, 3.18470245009348320806537757475, 5.00386198706595367086984655004, 5.69370182602562433079523799994, 7.24901748483705289097621979862, 7.68899614471290168864108519925, 8.979557837356359118952650746872, 9.545562058268836328979601409514, 10.81252226507216044774908645000
