| L(s) = 1 | + 4.15·2-s + 29.2·3-s − 14.7·4-s + 25·5-s + 121.·6-s + 42.5·7-s − 194.·8-s + 611.·9-s + 103.·10-s + 434.·11-s − 431.·12-s − 169·13-s + 176.·14-s + 730.·15-s − 334.·16-s − 424.·17-s + 2.53e3·18-s − 2.20e3·19-s − 368.·20-s + 1.24e3·21-s + 1.80e3·22-s − 1.17e3·23-s − 5.67e3·24-s + 625·25-s − 701.·26-s + 1.07e4·27-s − 627.·28-s + ⋯ |
| L(s) = 1 | + 0.734·2-s + 1.87·3-s − 0.460·4-s + 0.447·5-s + 1.37·6-s + 0.327·7-s − 1.07·8-s + 2.51·9-s + 0.328·10-s + 1.08·11-s − 0.864·12-s − 0.277·13-s + 0.240·14-s + 0.838·15-s − 0.326·16-s − 0.356·17-s + 1.84·18-s − 1.40·19-s − 0.206·20-s + 0.614·21-s + 0.794·22-s − 0.463·23-s − 2.01·24-s + 0.200·25-s − 0.203·26-s + 2.84·27-s − 0.151·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.135324521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.135324521\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 - 4.15T + 32T^{2} \) |
| 3 | \( 1 - 29.2T + 243T^{2} \) |
| 7 | \( 1 - 42.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 434.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 424.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.20e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.24e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.15e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.08e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.28e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.61e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.63e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.81e5T + 8.58e9T^{2} \) |
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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
−13.89450496212874654968972125559, −13.29462829749581063809751958005, −12.16606705923920002004296057714, −10.06549512252618447844242123930, −9.030341132353807470199419549886, −8.335415990075900695551635195073, −6.63052217674944975833931317805, −4.57768005018023951016534301971, −3.51372216342502787829085168784, −1.98381328955235191340079263652,
1.98381328955235191340079263652, 3.51372216342502787829085168784, 4.57768005018023951016534301971, 6.63052217674944975833931317805, 8.335415990075900695551635195073, 9.030341132353807470199419549886, 10.06549512252618447844242123930, 12.16606705923920002004296057714, 13.29462829749581063809751958005, 13.89450496212874654968972125559
