| L(s) = 1 | + (3.23 + 2.35i)2-s + (2.19 + 20.8i)3-s + (4.94 + 15.2i)4-s + (6.52 + 11.3i)5-s + (−41.9 + 72.5i)6-s + (167. + 35.6i)7-s + (−19.7 + 60.8i)8-s + (−191. + 40.7i)9-s + (−5.45 + 51.9i)10-s + (−232. + 258. i)11-s + (−306. + 136. i)12-s + (−333. − 148. i)13-s + (458. + 509. i)14-s + (−221. + 160. i)15-s + (−207. + 150. i)16-s + (−1.33e3 − 1.48e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.140 + 1.33i)3-s + (0.154 + 0.475i)4-s + (0.116 + 0.202i)5-s + (−0.475 + 0.823i)6-s + (1.29 + 0.275i)7-s + (−0.109 + 0.336i)8-s + (−0.788 + 0.167i)9-s + (−0.0172 + 0.164i)10-s + (−0.579 + 0.643i)11-s + (−0.613 + 0.273i)12-s + (−0.548 − 0.244i)13-s + (0.625 + 0.695i)14-s + (−0.253 + 0.184i)15-s + (−0.202 + 0.146i)16-s + (−1.12 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.963347 + 2.40469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.963347 + 2.40469i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 31 | \( 1 + (-5.03e3 - 1.79e3i)T \) |
| good | 3 | \( 1 + (-2.19 - 20.8i)T + (-237. + 50.5i)T^{2} \) |
| 5 | \( 1 + (-6.52 - 11.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-167. - 35.6i)T + (1.53e4 + 6.83e3i)T^{2} \) |
| 11 | \( 1 + (232. - 258. i)T + (-1.68e4 - 1.60e5i)T^{2} \) |
| 13 | \( 1 + (333. + 148. i)T + (2.48e5 + 2.75e5i)T^{2} \) |
| 17 | \( 1 + (1.33e3 + 1.48e3i)T + (-1.48e5 + 1.41e6i)T^{2} \) |
| 19 | \( 1 + (-1.29e3 + 578. i)T + (1.65e6 - 1.84e6i)T^{2} \) |
| 23 | \( 1 + (526. - 1.62e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.91e3 - 3.57e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 37 | \( 1 + (403. - 699. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (454. - 4.32e3i)T + (-1.13e8 - 2.40e7i)T^{2} \) |
| 43 | \( 1 + (-2.96e3 + 1.31e3i)T + (9.83e7 - 1.09e8i)T^{2} \) |
| 47 | \( 1 + (-1.55e4 + 1.12e4i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-4.01e3 + 854. i)T + (3.82e8 - 1.70e8i)T^{2} \) |
| 59 | \( 1 + (4.23e3 + 4.02e4i)T + (-6.99e8 + 1.48e8i)T^{2} \) |
| 61 | \( 1 + 1.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.65e4 - 4.60e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (4.50e4 - 9.57e3i)T + (1.64e9 - 7.33e8i)T^{2} \) |
| 73 | \( 1 + (-1.73e4 + 1.93e4i)T + (-2.16e8 - 2.06e9i)T^{2} \) |
| 79 | \( 1 + (3.01e4 + 3.34e4i)T + (-3.21e8 + 3.06e9i)T^{2} \) |
| 83 | \( 1 + (1.45e3 - 1.38e4i)T + (-3.85e9 - 8.18e8i)T^{2} \) |
| 89 | \( 1 + (-1.73e4 - 5.34e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (189. + 584. i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.60590010025453847617407757892, −13.74752354205567613935362713602, −12.08368303626366362108507497730, −11.01133203764929241843848926882, −9.864562036559407229148605658407, −8.571912918541682268835357764427, −7.17743505608411695973825255060, −5.09334660678109245407707955125, −4.63045993735891840553004570185, −2.70962335923653799964977031034,
1.10289423446653215017365377214, 2.34483946672678741788243503786, 4.56781013432519478387608488525, 6.09738746413102030637621834182, 7.52899958770249200195008665282, 8.525904278661404479923372847152, 10.51428067078534354423492118731, 11.60176619534288817542529345356, 12.56774228912880205699991658025, 13.56000900282241817499612597247
