| L(s) = 1 | + (−1.38 + 7.87i)2-s + (65.8 + 55.2i)3-s + (−60.1 − 21.8i)4-s + (133. − 48.4i)5-s + (−526. + 441. i)6-s + (558. + 968. i)7-s + (256 − 443. i)8-s + (902. + 5.11e3i)9-s + (196. + 1.11e3i)10-s + (3.43e3 − 5.95e3i)11-s + (−2.74e3 − 4.76e3i)12-s + (−6.05e3 + 5.08e3i)13-s + (−8.40e3 + 3.05e3i)14-s + (1.14e4 + 4.16e3i)15-s + (3.13e3 + 2.63e3i)16-s + (3.27e3 − 1.85e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (1.40 + 1.18i)3-s + (−0.469 − 0.171i)4-s + (0.476 − 0.173i)5-s + (−0.995 + 0.835i)6-s + (0.615 + 1.06i)7-s + (0.176 − 0.306i)8-s + (0.412 + 2.34i)9-s + (0.0622 + 0.352i)10-s + (0.779 − 1.34i)11-s + (−0.459 − 0.795i)12-s + (−0.764 + 0.641i)13-s + (−0.818 + 0.297i)14-s + (0.874 + 0.318i)15-s + (0.191 + 0.160i)16-s + (0.161 − 0.917i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.25268 + 2.48855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25268 + 2.48855i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 - 7.87i)T \) |
| 19 | \( 1 + (2.96e4 - 3.88e3i)T \) |
| good | 3 | \( 1 + (-65.8 - 55.2i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (-133. + 48.4i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-558. - 968. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.43e3 + 5.95e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (6.05e3 - 5.08e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-3.27e3 + 1.85e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-2.21e4 - 8.05e3i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (2.14e4 + 1.21e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (1.16e5 + 2.01e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.54e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.75e5 - 3.14e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-6.20e5 + 2.25e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-1.95e4 - 1.10e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (1.42e5 + 5.18e4i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (7.30e4 - 4.14e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-2.50e6 - 9.11e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (3.41e5 + 1.93e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (4.78e5 - 1.74e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (9.31e5 + 7.81e5i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (3.34e5 + 2.80e5i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-1.90e6 - 3.29e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-5.61e5 + 4.71e5i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-2.79e6 + 1.58e7i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−14.98664479068806722594061132178, −14.43893348213780935590070986533, −13.45278190226934584328754683941, −11.32800507519390515207256660489, −9.496393513868530886663163727457, −9.054203526283530967737373129470, −7.936138101031442559908946245625, −5.66663621015587656610197627504, −4.21899832157436301749209358113, −2.39757105330433128540842863107,
1.27485804848168326618318482228, 2.30882412245176286424104229374, 4.05715745319883662243511307734, 6.93693280770863796919821884989, 7.917269050670089446952380712894, 9.260347818893091114636568986587, 10.49039433038328856467690540577, 12.40934873753517015077577570615, 13.02816535469412274238235722954, 14.43245027120650472950058009749
