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
−15.32248787947839477107446491456, −14.87038916201874647062865954853, −12.75186158765420683405872156269, −11.37657783441804674546055899281, −10.31947166588550501908181901232, −9.028405859140042399135430423593, −8.010059479225446809764250612619, −5.96401395904713845464966432484, −4.59349237509896008746743897774, −2.12032709646715923900740708061,
0.49721526479677054067057521045, 1.87065010523945040910893248963, 4.30587988675810648854959694358, 6.85669863861108753783236788084, 7.48976605732246249369248624081, 9.066906412714606391141479477347, 10.74219992936793012263348471655, 11.49886343592225550863657628295, 13.05031963732181598266116468679, 13.92489374730626475650540367007