| L(s) = 1 | + (−7.51 + 2.73i)2-s + (7.10 − 40.2i)3-s + (49.0 − 41.1i)4-s + (−241. − 202. i)5-s + (56.8 + 322. i)6-s + (−369. − 640. i)7-s + (−256. + 443. i)8-s + (482. + 175. i)9-s + (2.36e3 + 861. i)10-s + (195. − 338. i)11-s + (−1.30e3 − 2.26e3i)12-s + (838. + 4.75e3i)13-s + (4.52e3 + 3.80e3i)14-s + (−9.87e3 + 8.28e3i)15-s + (711. − 4.03e3i)16-s + (−1.47e4 + 5.38e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.151 − 0.861i)3-s + (0.383 − 0.321i)4-s + (−0.863 − 0.724i)5-s + (0.107 + 0.609i)6-s + (−0.407 − 0.705i)7-s + (−0.176 + 0.306i)8-s + (0.220 + 0.0802i)9-s + (0.748 + 0.272i)10-s + (0.0442 − 0.0766i)11-s + (−0.218 − 0.378i)12-s + (0.105 + 0.600i)13-s + (0.441 + 0.370i)14-s + (−0.755 + 0.633i)15-s + (0.0434 − 0.246i)16-s + (−0.730 + 0.265i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0559198 + 0.167260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0559198 + 0.167260i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.51 - 2.73i)T \) |
| 19 | \( 1 + (2.56e4 - 1.53e4i)T \) |
| good | 3 | \( 1 + (-7.10 + 40.2i)T + (-2.05e3 - 747. i)T^{2} \) |
| 5 | \( 1 + (241. + 202. i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (369. + 640. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-195. + 338. i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-838. - 4.75e3i)T + (-5.89e7 + 2.14e7i)T^{2} \) |
| 17 | \( 1 + (1.47e4 - 5.38e3i)T + (3.14e8 - 2.63e8i)T^{2} \) |
| 23 | \( 1 + (6.85e4 - 5.75e4i)T + (5.91e8 - 3.35e9i)T^{2} \) |
| 29 | \( 1 + (2.83e4 + 1.03e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-1.03e5 - 1.78e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 2.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.17e4 + 1.79e5i)T + (-1.83e11 - 6.66e10i)T^{2} \) |
| 43 | \( 1 + (3.59e5 + 3.01e5i)T + (4.72e10 + 2.67e11i)T^{2} \) |
| 47 | \( 1 + (3.33e5 + 1.21e5i)T + (3.88e11 + 3.25e11i)T^{2} \) |
| 53 | \( 1 + (7.98e5 - 6.69e5i)T + (2.03e11 - 1.15e12i)T^{2} \) |
| 59 | \( 1 + (9.73e5 - 3.54e5i)T + (1.90e12 - 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-1.48e6 + 1.24e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (1.51e6 + 5.50e5i)T + (4.64e12 + 3.89e12i)T^{2} \) |
| 71 | \( 1 + (1.30e6 + 1.09e6i)T + (1.57e12 + 8.95e12i)T^{2} \) |
| 73 | \( 1 + (8.17e5 - 4.63e6i)T + (-1.03e13 - 3.77e12i)T^{2} \) |
| 79 | \( 1 + (-7.85e4 + 4.45e5i)T + (-1.80e13 - 6.56e12i)T^{2} \) |
| 83 | \( 1 + (4.60e6 + 7.97e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (1.37e6 + 7.81e6i)T + (-4.15e13 + 1.51e13i)T^{2} \) |
| 97 | \( 1 + (7.71e6 - 2.80e6i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−13.93753161373544305406450087514, −12.81172453720140904514073228869, −11.71732562703713108680093265944, −10.19573870450845261759165063540, −8.614585788991105480208157027862, −7.63808888590678180123949698321, −6.50568073410617079899123860243, −4.17759446319303934033513867381, −1.60228040620431960949110757141, −0.094013286943387819348158788779,
2.74433870951767970466636153654, 4.19318170476774969911078483617, 6.52821543466956261902081748523, 8.092315481077346974135216062779, 9.392315246621308850626875370722, 10.49208794880352532963263627882, 11.52699560112470132009748954821, 12.85154902980313111224829709093, 14.91647057627067445394362316530, 15.48049161478419741799790629353
