| L(s) = 1 | + (−52.4 + 19.0i)3-s + (31.0 − 175. i)5-s + (324. + 561. i)7-s + (709. − 594. i)9-s + (−3.80e3 + 6.59e3i)11-s + (−1.16e3 − 423. i)13-s + (1.73e3 + 9.81e3i)15-s + (−2.12e4 − 1.78e4i)17-s + (2.78e4 + 1.07e4i)19-s + (−2.77e4 − 2.32e4i)21-s + (−1.12e4 − 6.35e4i)23-s + (4.34e4 + 1.58e4i)25-s + (3.51e4 − 6.09e4i)27-s + (1.82e5 − 1.53e5i)29-s + (−9.21e3 − 1.59e4i)31-s + ⋯ |
| L(s) = 1 | + (−1.12 + 0.408i)3-s + (0.111 − 0.629i)5-s + (0.357 + 0.618i)7-s + (0.324 − 0.272i)9-s + (−0.862 + 1.49i)11-s + (−0.147 − 0.0535i)13-s + (0.132 + 0.751i)15-s + (−1.04 − 0.880i)17-s + (0.932 + 0.360i)19-s + (−0.652 − 0.547i)21-s + (−0.192 − 1.08i)23-s + (0.555 + 0.202i)25-s + (0.344 − 0.595i)27-s + (1.39 − 1.16i)29-s + (−0.0555 − 0.0961i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.487650 - 0.411049i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487650 - 0.411049i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.78e4 - 1.07e4i)T \) |
| good | 3 | \( 1 + (52.4 - 19.0i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-31.0 + 175. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-324. - 561. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.80e3 - 6.59e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.16e3 + 423. i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (2.12e4 + 1.78e4i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (1.12e4 + 6.35e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-1.82e5 + 1.53e5i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (9.21e3 + 1.59e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.76e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (5.54e5 - 2.01e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-9.65e4 + 5.47e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-8.41e4 + 7.05e4i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (6.19e4 + 3.51e5i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (1.75e6 + 1.47e6i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (3.14e5 + 1.78e6i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (-2.69e6 + 2.25e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (4.09e5 - 2.32e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-1.04e6 + 3.80e5i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-2.59e6 + 9.44e5i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-2.36e6 - 4.09e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-7.06e6 - 2.57e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (2.30e6 + 1.93e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.43323877844070008209927280408, −11.88321988227393607223377920462, −10.61826860709148534615426994141, −9.647063503868204041061867290991, −8.272788039550917401549689850137, −6.72642501199262075721287108439, −5.13850729892568412503762022262, −4.78902325227069001693186467788, −2.25061843375580432614309675009, −0.29208046302399114487056628079,
1.08923011365945072894636319531, 3.15826184189357083178195589272, 5.08162381063490703033341300066, 6.20288077710917134229324186885, 7.22318008037958472155858207198, 8.630278143559295335939337524836, 10.57732014100517857706202998715, 10.94483677721719613433079597348, 12.02871593379341675753735412568, 13.35493333764855165402195234181
