| L(s) = 1 | + (−7.51 − 2.73i)2-s + (−7.18 − 40.7i)3-s + (49.0 + 41.1i)4-s + (−24.6 + 20.6i)5-s + (−57.4 + 325. i)6-s + (764. − 1.32e3i)7-s + (−256. − 443. i)8-s + (448. − 163. i)9-s + (241. − 87.9i)10-s + (−523. − 907. i)11-s + (1.32e3 − 2.29e3i)12-s + (−1.06e3 + 6.06e3i)13-s + (−9.36e3 + 7.85e3i)14-s + (1.01e3 + 854. i)15-s + (711. + 4.03e3i)16-s + (−3.22e4 − 1.17e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−0.153 − 0.870i)3-s + (0.383 + 0.321i)4-s + (−0.0880 + 0.0739i)5-s + (−0.108 + 0.615i)6-s + (0.842 − 1.45i)7-s + (−0.176 − 0.306i)8-s + (0.204 − 0.0745i)9-s + (0.0764 − 0.0278i)10-s + (−0.118 − 0.205i)11-s + (0.221 − 0.382i)12-s + (−0.134 + 0.765i)13-s + (−0.912 + 0.765i)14-s + (0.0778 + 0.0653i)15-s + (0.0434 + 0.246i)16-s + (−1.59 − 0.580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.184i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0840425 - 0.904605i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0840425 - 0.904605i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (1.62e4 + 2.51e4i)T \) |
| good | 3 | \( 1 + (7.18 + 40.7i)T + (-2.05e3 + 747. i)T^{2} \) |
| 5 | \( 1 + (24.6 - 20.6i)T + (1.35e4 - 7.69e4i)T^{2} \) |
| 7 | \( 1 + (-764. + 1.32e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (523. + 907. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.06e3 - 6.06e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (3.22e4 + 1.17e4i)T + (3.14e8 + 2.63e8i)T^{2} \) |
| 23 | \( 1 + (-4.86e3 - 4.08e3i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (6.39e4 - 2.32e4i)T + (1.32e10 - 1.10e10i)T^{2} \) |
| 31 | \( 1 + (1.81e4 - 3.14e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 9.60e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (5.12e4 + 2.90e5i)T + (-1.83e11 + 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-7.12e5 + 5.97e5i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (4.92e5 - 1.79e5i)T + (3.88e11 - 3.25e11i)T^{2} \) |
| 53 | \( 1 + (-1.22e6 - 1.02e6i)T + (2.03e11 + 1.15e12i)T^{2} \) |
| 59 | \( 1 + (7.72e5 + 2.81e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (2.05e6 + 1.72e6i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-2.88e6 + 1.04e6i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (3.38e6 - 2.84e6i)T + (1.57e12 - 8.95e12i)T^{2} \) |
| 73 | \( 1 + (-2.81e5 - 1.59e6i)T + (-1.03e13 + 3.77e12i)T^{2} \) |
| 79 | \( 1 + (3.05e5 + 1.73e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-1.22e6 + 2.12e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-2.67e5 + 1.51e6i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (-1.07e7 - 3.92e6i)T + (6.18e13 + 5.19e13i)T^{2} \) |
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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
−13.92489374730626475650540367007, −13.05031963732181598266116468679, −11.49886343592225550863657628295, −10.74219992936793012263348471655, −9.066906412714606391141479477347, −7.48976605732246249369248624081, −6.85669863861108753783236788084, −4.30587988675810648854959694358, −1.87065010523945040910893248963, −0.49721526479677054067057521045,
2.12032709646715923900740708061, 4.59349237509896008746743897774, 5.96401395904713845464966432484, 8.010059479225446809764250612619, 9.028405859140042399135430423593, 10.31947166588550501908181901232, 11.37657783441804674546055899281, 12.75186158765420683405872156269, 14.87038916201874647062865954853, 15.32248787947839477107446491456
