| L(s) = 1 | + 5.93·2-s + 3.25·4-s + 25·5-s − 105.·7-s − 170.·8-s + 148.·10-s + 121·11-s + 124.·13-s − 626.·14-s − 1.11e3·16-s − 1.76e3·17-s + 1.99e3·19-s + 81.3·20-s + 718.·22-s + 2.54e3·23-s + 625·25-s + 737.·26-s − 343.·28-s + 320.·29-s + 8.55e3·31-s − 1.17e3·32-s − 1.04e4·34-s − 2.63e3·35-s + 4.12e3·37-s + 1.18e4·38-s − 4.26e3·40-s − 4.53e3·41-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.101·4-s + 0.447·5-s − 0.814·7-s − 0.942·8-s + 0.469·10-s + 0.301·11-s + 0.203·13-s − 0.854·14-s − 1.09·16-s − 1.48·17-s + 1.27·19-s + 0.0454·20-s + 0.316·22-s + 1.00·23-s + 0.200·25-s + 0.214·26-s − 0.0827·28-s + 0.0707·29-s + 1.59·31-s − 0.202·32-s − 1.55·34-s − 0.364·35-s + 0.495·37-s + 1.33·38-s − 0.421·40-s − 0.421·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.072045387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.072045387\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 5.93T + 32T^{2} \) |
| 7 | \( 1 + 105.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 124.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.76e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.99e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 320.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.53e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.37e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.17e4T + 8.58e9T^{2} \) |
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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
−10.06862564078563669568845737908, −9.289710336320112187466154236872, −8.564910738221870328834909613173, −6.96239600395792714250600294858, −6.33917415394029115880112675984, −5.36707306519487325413726335084, −4.45693052594600317700836379907, −3.39472064896941980051460272266, −2.49833647274847084199304133928, −0.75048504206904070385081125681,
0.75048504206904070385081125681, 2.49833647274847084199304133928, 3.39472064896941980051460272266, 4.45693052594600317700836379907, 5.36707306519487325413726335084, 6.33917415394029115880112675984, 6.96239600395792714250600294858, 8.564910738221870328834909613173, 9.289710336320112187466154236872, 10.06862564078563669568845737908
