| 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
−15.45968912107725946399973040688, −14.66737177322805064028902640206, −13.15370268977535430331269417676, −11.34898422294923459838765743765, −11.16350008620937992390002900776, −9.585942598829516291944572691854, −8.365606236580344684977310636784, −5.56368502833615515111061194034, −4.88031419779383961409441802820, −3.03604013751761493790976320341,
1.57179768240924143410804568316, 4.38972451338513699740910376044, 6.27788847359387842424190398579, 7.43101022700503879319727275017, 8.180210870669238208003020100637, 10.83112310776913767685711231750, 11.80676446953119248799426806895, 13.10099093517708970638552030493, 13.73192195578549533514210550691, 14.71540802547379748340822605565
