| 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 | \( 1 + (-5.59e3 - 1.49e3i)T + (7.66e7 + 4.42e7i)T^{2} \) |
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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
−13.62648525495034792406779363819, −12.61320066218056568331062765773, −11.89054849916950381110601494885, −10.83787707700594132114863010319, −9.315454948545530604580441801673, −7.83902233087435463019658052874, −5.95114577937715001867640917779, −5.43720540353178307140720382302, −3.99653481379844392452855451909, −3.35729310842594322923232594032,
1.08601419333857415222416860111, 3.06847930608583741877478169016, 4.04399631457696588384188113717, 6.01440180105221694478155427431, 6.72058550262266652238765369628, 7.62316126972796941612379853820, 10.26506461466310383031801757579, 11.38051085976654828768918636165, 11.95070071074314206207786192024, 13.11988682006527384095582983800
