| L(s) = 1 | + (−4.49 − 3.26i)2-s + (6.79 + 20.9i)3-s + (−0.349 − 1.07i)4-s + (−34.6 − 43.8i)5-s + (37.7 − 116. i)6-s − 153.·7-s + (−56.8 + 175. i)8-s + (−194. + 141. i)9-s + (12.6 + 310. i)10-s + (−431. − 313. i)11-s + (20.1 − 14.6i)12-s + (−177. + 129. i)13-s + (689. + 501. i)14-s + (681. − 1.02e3i)15-s + (798. − 579. i)16-s + (273. − 841. i)17-s + ⋯ |
| L(s) = 1 | + (−0.794 − 0.577i)2-s + (0.435 + 1.34i)3-s + (−0.0109 − 0.0336i)4-s + (−0.620 − 0.784i)5-s + (0.427 − 1.31i)6-s − 1.18·7-s + (−0.314 + 0.967i)8-s + (−0.799 + 0.580i)9-s + (0.0400 + 0.981i)10-s + (−1.07 − 0.781i)11-s + (0.0403 − 0.0293i)12-s + (−0.291 + 0.212i)13-s + (0.940 + 0.683i)14-s + (0.781 − 1.17i)15-s + (0.779 − 0.566i)16-s + (0.229 − 0.706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0441i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.00106649 + 0.0483372i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00106649 + 0.0483372i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (34.6 + 43.8i)T \) |
| good | 2 | \( 1 + (4.49 + 3.26i)T + (9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (-6.79 - 20.9i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + 153.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (431. + 313. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (177. - 129. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-273. + 841. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (488. - 1.50e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-2.76e3 - 2.00e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (2.59e3 + 7.99e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (182. - 562. i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (5.74e3 - 4.17e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (7.61e3 - 5.53e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.44e3 + 4.44e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (7.86e3 + 2.41e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.07e4 + 1.50e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.86e4 - 1.35e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (254. - 782. i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.25e4 + 3.87e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (4.91e4 + 3.57e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (4.34e3 + 1.33e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.01e4 - 9.28e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (1.03e4 + 7.52e3i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (1.17e3 + 3.61e3i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−16.02644960254263514966216155780, −15.06448263613416741448776682782, −13.31667081439455082037194802513, −11.56383456798136090892763510745, −10.15974916858280017496254480581, −9.434591198087909016481176218873, −8.279042233357114135390339991986, −5.21891835129237145693773761506, −3.30498001968618346612090483244, −0.03833105366392363845684923482,
2.97274073893901942192236882984, 6.81109451591909498149277841500, 7.31090298198331205227264329236, 8.629448121183820427536387115520, 10.34122439335437936225240612781, 12.55749570499386106491564794940, 13.03336605876243368169987643792, 14.91549818408668142781986033902, 15.99393431665735116796978102703, 17.45153605125290296610418585588
