| L(s) = 1 | + (0.385 + 0.140i)3-s + (3.36 + 19.1i)5-s + (63.2 − 109. i)7-s + (−186. − 156. i)9-s + (246. + 426. i)11-s + (587. − 213. i)13-s + (−1.38 + 7.84i)15-s + (1.22e3 − 1.02e3i)17-s + (102. − 1.57e3i)19-s + (39.7 − 33.3i)21-s + (−31.0 + 176. i)23-s + (2.58e3 − 940. i)25-s + (−99.6 − 172. i)27-s + (4.20e3 + 3.52e3i)29-s + (5.12e3 − 8.87e3i)31-s + ⋯ |
| L(s) = 1 | + (0.0247 + 0.00900i)3-s + (0.0602 + 0.341i)5-s + (0.487 − 0.844i)7-s + (−0.765 − 0.642i)9-s + (0.613 + 1.06i)11-s + (0.964 − 0.350i)13-s + (−0.00158 + 0.00899i)15-s + (1.02 − 0.861i)17-s + (0.0650 − 0.997i)19-s + (0.0196 − 0.0164i)21-s + (−0.0122 + 0.0695i)23-s + (0.826 − 0.300i)25-s + (−0.0263 − 0.0455i)27-s + (0.928 + 0.778i)29-s + (0.957 − 1.65i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.81210 - 0.564339i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81210 - 0.564339i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-102. + 1.57e3i)T \) |
| good | 3 | \( 1 + (-0.385 - 0.140i)T + (186. + 156. i)T^{2} \) |
| 5 | \( 1 + (-3.36 - 19.1i)T + (-2.93e3 + 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-63.2 + 109. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-246. - 426. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-587. + 213. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-1.22e3 + 1.02e3i)T + (2.46e5 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (31.0 - 176. i)T + (-6.04e6 - 2.20e6i)T^{2} \) |
| 29 | \( 1 + (-4.20e3 - 3.52e3i)T + (3.56e6 + 2.01e7i)T^{2} \) |
| 31 | \( 1 + (-5.12e3 + 8.87e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-2.36e3 - 862. i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-2.97e3 - 1.69e4i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (2.02e4 + 1.69e4i)T + (3.98e7 + 2.25e8i)T^{2} \) |
| 53 | \( 1 + (5.15e3 - 2.92e4i)T + (-3.92e8 - 1.43e8i)T^{2} \) |
| 59 | \( 1 + (6.13e3 - 5.14e3i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (3.73e3 - 2.11e4i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (3.05e4 + 2.56e4i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (5.80e3 + 3.29e4i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 + (1.89e4 + 6.89e3i)T + (1.58e9 + 1.33e9i)T^{2} \) |
| 79 | \( 1 + (-2.90e4 - 1.05e4i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-2.95e4 + 5.11e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (9.13e4 - 3.32e4i)T + (4.27e9 - 3.58e9i)T^{2} \) |
| 97 | \( 1 + (-5.22e4 + 4.38e4i)T + (1.49e9 - 8.45e9i)T^{2} \) |
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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
−13.64564527824755814285227950068, −12.19498778053289478896711071642, −11.24312521662112934046350546921, −10.11272806160562348731234433396, −8.889973071789147471102492648620, −7.49961180908374501164341647149, −6.37345552745776912346513677182, −4.68102603315984644033101032632, −3.13099499733192427642872490493, −0.980847807075744502069919859591,
1.45692182510985821699703694967, 3.36591190764052701607507748526, 5.25000472155792282839479614702, 6.23465787040684533412844416040, 8.378770375065434642917220013939, 8.615649878441788759343846989027, 10.41551650276447796671112659736, 11.52210730306739184729732226478, 12.38619121983392217370478728297, 13.87874384217693427826787884358
