| L(s) = 1 | + (7.05 − 1.89i)2-s + (−3.10 − 11.6i)3-s + (32.3 − 18.6i)4-s + (−19.2 + 15.8i)5-s + (−43.9 − 76.0i)6-s + (−48.0 − 48.0i)7-s + (110. − 110. i)8-s + (−54.8 + 31.6i)9-s + (−106. + 148. i)10-s − 17.4·11-s + (−317. − 317. i)12-s + (279. + 74.9i)13-s + (−429. − 248. i)14-s + (244. + 174. i)15-s + (272. − 471. i)16-s + (251. − 67.4i)17-s + ⋯ |
| L(s) = 1 | + (1.76 − 0.472i)2-s + (−0.345 − 1.28i)3-s + (2.02 − 1.16i)4-s + (−0.771 + 0.635i)5-s + (−1.21 − 2.11i)6-s + (−0.980 − 0.980i)7-s + (1.72 − 1.72i)8-s + (−0.677 + 0.391i)9-s + (−1.06 + 1.48i)10-s − 0.143·11-s + (−2.20 − 2.20i)12-s + (1.65 + 0.443i)13-s + (−2.19 − 1.26i)14-s + (1.08 + 0.775i)15-s + (1.06 − 1.84i)16-s + (0.870 − 0.233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.42268 - 3.22573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42268 - 3.22573i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (19.2 - 15.8i)T \) |
| 19 | \( 1 + (-329. + 147. i)T \) |
| good | 2 | \( 1 + (-7.05 + 1.89i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (3.10 + 11.6i)T + (-70.1 + 40.5i)T^{2} \) |
| 7 | \( 1 + (48.0 + 48.0i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 17.4T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-279. - 74.9i)T + (2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (-251. + 67.4i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 23 | \( 1 + (403. + 108. i)T + (2.42e5 + 1.39e5i)T^{2} \) |
| 29 | \( 1 + (403. - 233. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 - 139.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-1.45e3 - 1.45e3i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + (198. - 344. i)T + (-1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-539. - 2.01e3i)T + (-2.96e6 + 1.70e6i)T^{2} \) |
| 47 | \( 1 + (-426. + 1.59e3i)T + (-4.22e6 - 2.43e6i)T^{2} \) |
| 53 | \( 1 + (-2.73e3 - 733. i)T + (6.83e6 + 3.94e6i)T^{2} \) |
| 59 | \( 1 + (5.60e3 + 3.23e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.20e3 - 3.82e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (1.62e3 - 6.07e3i)T + (-1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 + (626. - 1.08e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-500. - 1.86e3i)T + (-2.45e7 + 1.41e7i)T^{2} \) |
| 79 | \( 1 + (6.89e3 + 3.97e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-7.60e3 + 7.60e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (-4.79e3 + 2.76e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-5.59e3 + 1.49e3i)T + (7.66e7 - 4.42e7i)T^{2} \) |
| 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
−13.11988682006527384095582983800, −11.95070071074314206207786192024, −11.38051085976654828768918636165, −10.26506461466310383031801757579, −7.62316126972796941612379853820, −6.72058550262266652238765369628, −6.01440180105221694478155427431, −4.04399631457696588384188113717, −3.06847930608583741877478169016, −1.08601419333857415222416860111,
3.35729310842594322923232594032, 3.99653481379844392452855451909, 5.43720540353178307140720382302, 5.95114577937715001867640917779, 7.83902233087435463019658052874, 9.315454948545530604580441801673, 10.83787707700594132114863010319, 11.89054849916950381110601494885, 12.61320066218056568331062765773, 13.62648525495034792406779363819
