| L(s) = 1 | + (−1.86 − 0.295i)2-s + (−2.19 − 4.30i)3-s + (−0.405 − 0.131i)4-s + (4.99 + 0.209i)5-s + (2.82 + 8.69i)6-s + (−3.57 − 3.57i)7-s + (7.45 + 3.79i)8-s + (−8.44 + 11.6i)9-s + (−9.26 − 1.86i)10-s + (11.7 − 8.53i)11-s + (0.322 + 2.03i)12-s + (1.48 − 0.234i)13-s + (5.61 + 7.72i)14-s + (−10.0 − 21.9i)15-s + (−11.4 − 8.29i)16-s + (0.980 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.933 − 0.147i)2-s + (−0.731 − 1.43i)3-s + (−0.101 − 0.0329i)4-s + (0.999 + 0.0419i)5-s + (0.470 + 1.44i)6-s + (−0.510 − 0.510i)7-s + (0.931 + 0.474i)8-s + (−0.938 + 1.29i)9-s + (−0.926 − 0.186i)10-s + (1.06 − 0.776i)11-s + (0.0268 + 0.169i)12-s + (0.114 − 0.0180i)13-s + (0.400 + 0.551i)14-s + (−0.670 − 1.46i)15-s + (−0.713 − 0.518i)16-s + (0.0576 − 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.309141 - 0.407301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.309141 - 0.407301i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.99 - 0.209i)T \) |
| good | 2 | \( 1 + (1.86 + 0.295i)T + (3.80 + 1.23i)T^{2} \) |
| 3 | \( 1 + (2.19 + 4.30i)T + (-5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (3.57 + 3.57i)T + 49iT^{2} \) |
| 11 | \( 1 + (-11.7 + 8.53i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (-1.48 + 0.234i)T + (160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-0.980 + 1.92i)T + (-169. - 233. i)T^{2} \) |
| 19 | \( 1 + (0.665 - 0.216i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (5.44 - 34.3i)T + (-503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-23.5 - 7.64i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (0.269 + 0.830i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-3.63 - 22.9i)T + (-1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (17.9 + 13.0i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-7.47 + 7.47i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-69.0 + 35.1i)T + (1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-11.3 - 22.3i)T + (-1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (38.5 - 53.0i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (71.2 - 51.7i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (40.4 - 79.4i)T + (-2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (8.73 - 26.8i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-13.1 + 82.8i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (102. + 33.3i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (22.5 + 11.4i)T + (4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-21.8 - 30.0i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-53.1 + 27.0i)T + (5.53e3 - 7.61e3i)T^{2} \) |
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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
−17.29768694355859080949111280915, −16.70272844165463050815972835167, −13.89301999071141002478697131779, −13.39061968369439524946298368842, −11.76982003098744870589957220963, −10.36025834638082769190588553111, −8.932033104745545337016408762169, −7.19879687598290177044992151781, −5.89142886253702718005711281505, −1.24150016808955104390600430118,
4.51946622826956937814092887161, 6.30871362946539354612327930223, 8.985233585838164923059370551932, 9.700799445401889793577417582801, 10.61733643626499469224211977502, 12.45854313425682408749401634404, 14.30904902665665503317300903512, 15.76129956690790477499231483083, 16.82202957936817623566054251134, 17.39045793582760790121491633947
