| L(s) = 1 | + (−1.46 + 1.36i)2-s + (6.60 − 0.995i)3-s + (0.298 − 3.98i)4-s + (−1.48 + 3.78i)5-s + (−8.32 + 10.4i)6-s + (2.76 + 18.3i)7-s + (4.98 + 6.25i)8-s + (16.7 − 5.18i)9-s + (−2.97 − 7.57i)10-s + (57.4 + 17.7i)11-s + (−1.99 − 26.6i)12-s + (−6.81 − 29.8i)13-s + (−28.9 − 23.0i)14-s + (−6.04 + 26.4i)15-s + (−15.8 − 2.38i)16-s + (29.9 − 20.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.518 + 0.480i)2-s + (1.27 − 0.191i)3-s + (0.0373 − 0.498i)4-s + (−0.132 + 0.338i)5-s + (−0.566 + 0.710i)6-s + (0.149 + 0.988i)7-s + (0.220 + 0.276i)8-s + (0.621 − 0.191i)9-s + (−0.0939 − 0.239i)10-s + (1.57 + 0.486i)11-s + (−0.0480 − 0.640i)12-s + (−0.145 − 0.637i)13-s + (−0.552 − 0.440i)14-s + (−0.104 + 0.455i)15-s + (−0.247 − 0.0372i)16-s + (0.427 − 0.291i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.808i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.589 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.63146 + 0.829680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63146 + 0.829680i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.46 - 1.36i)T \) |
| 7 | \( 1 + (-2.76 - 18.3i)T \) |
| good | 3 | \( 1 + (-6.60 + 0.995i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (1.48 - 3.78i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (-57.4 - 17.7i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (6.81 + 29.8i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (-29.9 + 20.4i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (-52.1 - 90.2i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (55.0 + 37.5i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (161. + 77.6i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-14.6 + 25.3i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (20.6 + 275. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (191. + 240. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (0.461 - 0.578i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-17.0 + 15.8i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (-44.5 + 594. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (153. + 391. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (0.223 + 2.98i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (438. - 759. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-582. + 280. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-671. - 623. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (300. + 519. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-244. + 1.07e3i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (740. - 228. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 + 869.T + 9.12e5T^{2} \) |
| 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
−14.10697491235731942887929867117, −12.58042293582282741785038421258, −11.51214598106481931479254733956, −9.834860265403718846913757770910, −9.072250283954553800833602436706, −8.132271962160836639736569598765, −7.13170427875127181189026564803, −5.66181488600716335610121369968, −3.54540025437510300946677419425, −1.92576507309210200342714886807,
1.32871400760814317696588395328, 3.26809955320932827385264025722, 4.30060542866010033989302719957, 6.82374384539244040634905015936, 8.030593033423826775644945531833, 9.013270133702284952726676877691, 9.679649261546183123687797909602, 11.08699742870644050335271642566, 12.08147170514177843353538400212, 13.57851703235337553914800037042
