| L(s) = 1 | + 9.61·2-s + 28.9·3-s + 60.4·4-s − 25·5-s + 278.·6-s − 189.·7-s + 273.·8-s + 596.·9-s − 240.·10-s − 566.·11-s + 1.75e3·12-s + 169·13-s − 1.82e3·14-s − 724.·15-s + 698.·16-s − 686.·17-s + 5.73e3·18-s + 125.·19-s − 1.51e3·20-s − 5.50e3·21-s − 5.45e3·22-s + 3.48e3·23-s + 7.93e3·24-s + 625·25-s + 1.62e3·26-s + 1.02e4·27-s − 1.14e4·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.85·3-s + 1.89·4-s − 0.447·5-s + 3.16·6-s − 1.46·7-s + 1.51·8-s + 2.45·9-s − 0.760·10-s − 1.41·11-s + 3.51·12-s + 0.277·13-s − 2.49·14-s − 0.831·15-s + 0.682·16-s − 0.576·17-s + 4.17·18-s + 0.0799·19-s − 0.845·20-s − 2.72·21-s − 2.40·22-s + 1.37·23-s + 2.81·24-s + 0.200·25-s + 0.471·26-s + 2.70·27-s − 2.76·28-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
| \(L(3)\) |
\(\approx\) |
\(6.122525079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.122525079\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 9.61T + 32T^{2} \) |
| 3 | \( 1 - 28.9T + 243T^{2} \) |
| 7 | \( 1 + 189.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 566.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 686.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 125.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.48e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.90e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.46e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.74e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.22e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.93e4T + 8.58e9T^{2} \) |
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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
−13.54342443405488051942266457269, −13.28307325529107633744530898325, −12.42439402243904708233274919269, −10.54866254812943793415526374721, −9.157346877239899324079010772461, −7.71964684036142276163564233258, −6.55139543087632280872520071166, −4.57601602086525222392157661926, −3.24015045185397993896592956832, −2.70551452962236041353648295606,
2.70551452962236041353648295606, 3.24015045185397993896592956832, 4.57601602086525222392157661926, 6.55139543087632280872520071166, 7.71964684036142276163564233258, 9.157346877239899324079010772461, 10.54866254812943793415526374721, 12.42439402243904708233274919269, 13.28307325529107633744530898325, 13.54342443405488051942266457269
