| L(s) = 1 | + (−1.48 + 0.483i)2-s + (−4.16 − 5.73i)3-s + (−4.49 + 3.26i)4-s + (−5.33 − 9.82i)5-s + (8.96 + 6.51i)6-s − 1.18i·7-s + (12.4 − 17.1i)8-s + (−7.16 + 22.0i)9-s + (12.6 + 12.0i)10-s + (7.19 + 22.1i)11-s + (37.3 + 12.1i)12-s + (−34.0 − 11.0i)13-s + (0.571 + 1.76i)14-s + (−34.1 + 71.4i)15-s + (3.46 − 10.6i)16-s + (76.1 − 104. i)17-s + ⋯ |
| L(s) = 1 | + (−0.526 + 0.170i)2-s + (−0.801 − 1.10i)3-s + (−0.561 + 0.407i)4-s + (−0.477 − 0.878i)5-s + (0.610 + 0.443i)6-s − 0.0638i·7-s + (0.550 − 0.758i)8-s + (−0.265 + 0.816i)9-s + (0.401 + 0.380i)10-s + (0.197 + 0.607i)11-s + (0.899 + 0.292i)12-s + (−0.726 − 0.235i)13-s + (0.0109 + 0.0336i)14-s + (−0.587 + 1.23i)15-s + (0.0541 − 0.166i)16-s + (1.08 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.146442 - 0.375757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.146442 - 0.375757i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (5.33 + 9.82i)T \) |
| good | 2 | \( 1 + (1.48 - 0.483i)T + (6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (4.16 + 5.73i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + 1.18iT - 343T^{2} \) |
| 11 | \( 1 + (-7.19 - 22.1i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (34.0 + 11.0i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-76.1 + 104. i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (85.6 + 62.2i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (71.8 - 23.3i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-216. + 157. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-192. - 139. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (166. + 54.0i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-6.36 + 19.5i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 131. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (101. + 139. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (19.3 + 26.6i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-148. + 457. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (174. + 537. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-198. + 272. i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (170. - 124. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-452. + 147. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (792. - 575. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-126. + 174. i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (120. + 369. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-325. - 447. i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
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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
−17.13776904661268822913077996895, −15.86161225176820261299604400697, −13.73752906222098710539795246082, −12.48926056891873708460987028247, −11.95623201406793135484864393222, −9.721147517693740991011947156629, −8.137240225655853970265351289894, −7.03242425907944539267235472169, −4.82758309220013362482739711205, −0.55404364337523548384309906463,
4.18314280638456821655115893299, 5.94238037812499746843436501066, 8.320779067002155009401490249261, 10.17333003970632348157610295457, 10.53372121782180993744629444954, 11.98599822213015579548955244237, 14.23864902363259903673874714679, 15.13154097444156826278368075699, 16.56795334406258846295974065594, 17.43441799332395022996735590339
