| L(s) = 1 | + (1.68 + 1.08i)2-s + (−1.15 − 8.06i)3-s + (1.66 + 3.63i)4-s + (−1.44 − 0.424i)5-s + (6.77 − 14.8i)6-s + (22.1 − 25.6i)7-s + (−1.13 + 7.91i)8-s + (−37.8 + 11.1i)9-s + (−1.97 − 2.27i)10-s + (−24.3 + 15.6i)11-s + (27.4 − 17.6i)12-s + (41.8 + 48.2i)13-s + (65.0 − 19.0i)14-s + (−1.74 + 12.1i)15-s + (−10.4 + 12.0i)16-s + (−30.0 + 65.8i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.223 − 1.55i)3-s + (0.207 + 0.454i)4-s + (−0.129 − 0.0379i)5-s + (0.460 − 1.00i)6-s + (1.19 − 1.38i)7-s + (−0.0503 + 0.349i)8-s + (−1.40 + 0.411i)9-s + (−0.0623 − 0.0719i)10-s + (−0.666 + 0.428i)11-s + (0.659 − 0.423i)12-s + (0.892 + 1.03i)13-s + (1.24 − 0.364i)14-s + (−0.0300 + 0.209i)15-s + (−0.163 + 0.188i)16-s + (−0.429 + 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.56122 - 0.785645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56122 - 0.785645i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 - 1.08i)T \) |
| 23 | \( 1 + (-108. + 18.6i)T \) |
| good | 3 | \( 1 + (1.15 + 8.06i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (1.44 + 0.424i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-22.1 + 25.6i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (24.3 - 15.6i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-41.8 - 48.2i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (30.0 - 65.8i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-23.7 - 51.9i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-36.3 + 79.5i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (11.3 - 79.2i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-298. + 87.6i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (67.2 + 19.7i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-14.3 - 99.4i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (414. - 478. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-205. - 236. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (45.0 - 313. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-24.8 - 15.9i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (159. + 102. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (464. + 1.01e3i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (544. + 627. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-546. + 160. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-81.9 - 570. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-355. - 104. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−14.71540802547379748340822605565, −13.73192195578549533514210550691, −13.10099093517708970638552030493, −11.80676446953119248799426806895, −10.83112310776913767685711231750, −8.180210870669238208003020100637, −7.43101022700503879319727275017, −6.27788847359387842424190398579, −4.38972451338513699740910376044, −1.57179768240924143410804568316,
3.03604013751761493790976320341, 4.88031419779383961409441802820, 5.56368502833615515111061194034, 8.365606236580344684977310636784, 9.585942598829516291944572691854, 11.16350008620937992390002900776, 11.34898422294923459838765743765, 13.15370268977535430331269417676, 14.66737177322805064028902640206, 15.45968912107725946399973040688
