| L(s) = 1 | − 11.2·2-s − 19.1·3-s + 94.0·4-s + 215.·6-s − 70.9·7-s − 696.·8-s + 125.·9-s − 161.·11-s − 1.80e3·12-s − 169·13-s + 796.·14-s + 4.80e3·16-s + 121.·17-s − 1.40e3·18-s − 3.11e3·19-s + 1.36e3·21-s + 1.80e3·22-s + 3.09e3·23-s + 1.33e4·24-s + 1.89e3·26-s + 2.25e3·27-s − 6.67e3·28-s + 3.17e3·29-s − 3.51e3·31-s − 3.16e4·32-s + 3.09e3·33-s − 1.36e3·34-s + ⋯ |
| L(s) = 1 | − 1.98·2-s − 1.23·3-s + 2.93·4-s + 2.44·6-s − 0.547·7-s − 3.84·8-s + 0.515·9-s − 0.401·11-s − 3.61·12-s − 0.277·13-s + 1.08·14-s + 4.69·16-s + 0.101·17-s − 1.02·18-s − 1.97·19-s + 0.674·21-s + 0.796·22-s + 1.22·23-s + 4.73·24-s + 0.550·26-s + 0.596·27-s − 1.60·28-s + 0.700·29-s − 0.656·31-s − 5.46·32-s + 0.493·33-s − 0.202·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.07815943346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07815943346\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169T \) |
| good | 2 | \( 1 + 11.2T + 32T^{2} \) |
| 3 | \( 1 + 19.1T + 243T^{2} \) |
| 7 | \( 1 + 70.9T + 1.68e4T^{2} \) |
| 11 | \( 1 + 161.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 121.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 3.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.51e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.95e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.63e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.31e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.25e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.12e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 634.T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
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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
−10.61168130047549516156901430968, −10.02747942224830979003685566544, −8.920125922132084143809839540117, −8.141249793289281581574885074104, −6.77319746609884665747931294484, −6.50692780307172948301081730835, −5.24024338354429699051157021898, −3.01061358536256109870286322813, −1.63390496731143236389025177829, −0.21735119812803655214682393124,
0.21735119812803655214682393124, 1.63390496731143236389025177829, 3.01061358536256109870286322813, 5.24024338354429699051157021898, 6.50692780307172948301081730835, 6.77319746609884665747931294484, 8.141249793289281581574885074104, 8.920125922132084143809839540117, 10.02747942224830979003685566544, 10.61168130047549516156901430968
