| L(s) = 1 | + 4·2-s − 14.3·3-s + 16·4-s + 25·5-s − 57.2·6-s + 49·7-s + 64·8-s − 38.5·9-s + 100·10-s + 425.·11-s − 228.·12-s + 399.·13-s + 196·14-s − 357.·15-s + 256·16-s + 1.75e3·17-s − 154.·18-s + 2.87e3·19-s + 400·20-s − 700.·21-s + 1.70e3·22-s − 2.31e3·23-s − 915.·24-s + 625·25-s + 1.59e3·26-s + 4.02e3·27-s + 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.917·3-s + 0.5·4-s + 0.447·5-s − 0.648·6-s + 0.377·7-s + 0.353·8-s − 0.158·9-s + 0.316·10-s + 1.06·11-s − 0.458·12-s + 0.655·13-s + 0.267·14-s − 0.410·15-s + 0.250·16-s + 1.46·17-s − 0.112·18-s + 1.82·19-s + 0.223·20-s − 0.346·21-s + 0.750·22-s − 0.911·23-s − 0.324·24-s + 0.200·25-s + 0.463·26-s + 1.06·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.365222516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.365222516\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 25T \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 + 14.3T + 243T^{2} \) |
| 11 | \( 1 - 425.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 399.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.75e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.87e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.89e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.31e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.65e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.24e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.16e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.21e4T + 8.58e9T^{2} \) |
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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
−13.94833495074810686294813597795, −12.36956813207223721330526075829, −11.69409130232467703355032397723, −10.70357815259358497715109754637, −9.289018081123149098751154123694, −7.49446706262649018256359447616, −6.01388239806070753905759773399, −5.33008247810111230407124841199, −3.56482832318045962041040716571, −1.31364771836085385253028848026,
1.31364771836085385253028848026, 3.56482832318045962041040716571, 5.33008247810111230407124841199, 6.01388239806070753905759773399, 7.49446706262649018256359447616, 9.289018081123149098751154123694, 10.70357815259358497715109754637, 11.69409130232467703355032397723, 12.36956813207223721330526075829, 13.94833495074810686294813597795
