| L(s) = 1 | + (5.56 − 4.04i)2-s + (−7.98 + 24.5i)3-s + (4.72 − 14.5i)4-s + (52.9 + 18.0i)5-s + (54.9 + 169. i)6-s − 5.99·7-s + (35.5 + 109. i)8-s + (−344. − 250. i)9-s + (367. − 113. i)10-s + (460. − 334. i)11-s + (319. + 232. i)12-s + (−694. − 504. i)13-s + (−33.3 + 24.2i)14-s + (−866. + 1.15e3i)15-s + (1.03e3 + 752. i)16-s + (282. + 869. i)17-s + ⋯ |
| L(s) = 1 | + (0.983 − 0.714i)2-s + (−0.512 + 1.57i)3-s + (0.147 − 0.454i)4-s + (0.946 + 0.322i)5-s + (0.623 + 1.91i)6-s − 0.0462·7-s + (0.196 + 0.603i)8-s + (−1.41 − 1.02i)9-s + (1.16 − 0.358i)10-s + (1.14 − 0.834i)11-s + (0.641 + 0.465i)12-s + (−1.14 − 0.828i)13-s + (−0.0454 + 0.0330i)14-s + (−0.994 + 1.32i)15-s + (1.01 + 0.734i)16-s + (0.237 + 0.730i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.07388 + 0.786476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07388 + 0.786476i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-52.9 - 18.0i)T \) |
| good | 2 | \( 1 + (-5.56 + 4.04i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (7.98 - 24.5i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 + 5.99T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-460. + 334. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (694. + 504. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-282. - 869. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (482. + 1.48e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-2.35e3 + 1.70e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (308. - 949. i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.59e3 + 4.90e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (477. + 346. i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (4.42e3 + 3.21e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 9.77e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (6.14e3 - 1.89e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.96e3 - 6.03e3i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.39e4 + 1.01e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (3.66e4 - 2.66e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.17e4 - 3.63e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.48e4 + 7.65e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.62e4 - 1.90e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.20e4 - 3.71e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (2.27e4 + 7.01e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (1.45e4 - 1.06e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.82e4 + 1.48e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−16.87371863902144571660433309039, −15.04009043829937391151495131613, −14.31315201838720502543914736246, −12.80346207584355285670532341392, −11.29634415633896618541030022843, −10.46212559074088717526279878125, −9.155111253793616534846139026065, −5.89176565347056311750968037256, −4.64506982984257347690706393654, −3.09297442229424304551684613223,
1.57117787663144773487028693162, 5.05996684782775501468976180138, 6.43399708852250695287153510743, 7.21431723659912624546973741057, 9.580719298594063449925306177215, 12.00064290322914062401722956876, 12.74163608440906105213776594897, 13.87931199131177695306459660146, 14.59912128937249777700493640632, 16.68200176532628007353097643699
