| L(s) = 1 | + (−0.267 − 1.51i)3-s + (−1.63 + 1.37i)5-s + (13.0 − 22.6i)7-s + (23.1 − 8.42i)9-s + (−11.3 − 19.7i)11-s + (4.34 − 24.6i)13-s + (2.51 + 2.11i)15-s + (−30.9 − 11.2i)17-s + (70.1 − 43.9i)19-s + (−37.8 − 13.7i)21-s + (113. + 95.2i)23-s + (−20.9 + 118. i)25-s + (−39.7 − 68.9i)27-s + (−205. + 74.8i)29-s + (−38.0 + 65.9i)31-s + ⋯ |
| L(s) = 1 | + (−0.0515 − 0.292i)3-s + (−0.146 + 0.122i)5-s + (0.705 − 1.22i)7-s + (0.857 − 0.311i)9-s + (−0.311 − 0.540i)11-s + (0.0927 − 0.526i)13-s + (0.0433 + 0.0363i)15-s + (−0.442 − 0.160i)17-s + (0.847 − 0.530i)19-s + (−0.393 − 0.143i)21-s + (1.02 + 0.863i)23-s + (−0.167 + 0.949i)25-s + (−0.283 − 0.491i)27-s + (−1.31 + 0.479i)29-s + (−0.220 + 0.382i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.30933 - 0.758473i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30933 - 0.758473i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-70.1 + 43.9i)T \) |
| good | 3 | \( 1 + (0.267 + 1.51i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (1.63 - 1.37i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (-13.0 + 22.6i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (11.3 + 19.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-4.34 + 24.6i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (30.9 + 11.2i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-113. - 95.2i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (205. - 74.8i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (38.0 - 65.9i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 76.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-56.3 - 319. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-4.61 + 3.86i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-285. + 103. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (76.7 + 64.4i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-280. - 102. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (16.1 + 13.5i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-658. + 239. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (35.9 - 30.2i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (58.8 + 333. i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (135. + 768. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (553. - 959. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (183. - 1.03e3i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (-1.68e3 - 612. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−13.60619508165908979855941181738, −13.02777865778315893489191166675, −11.43751897018409076761899410161, −10.69608171069576555544184126390, −9.345845184607329981537683380177, −7.71325122263652129906631061271, −7.03652249827814077659226777987, −5.16250327783781454786534042607, −3.60793456414454716338030714454, −1.12185441601549565166430806642,
2.09068707042312286050312170542, 4.34217553979304280705479738223, 5.52947751908951453655556018400, 7.25627593313366020429826400279, 8.550089819631386297171874883762, 9.649796367280924459556293802771, 10.93382126680932612464430928523, 12.03133460815765810316547946202, 12.95246086831705136184952691971, 14.37277084767169498571812346174
