| L(s) = 1 | − 2.66i·3-s + 0.750i·7-s − 4.09·9-s − 4.57·11-s − 3.84i·13-s + 7.32i·17-s − 6.57·19-s + 2.00·21-s + i·23-s + 2.91i·27-s + 7.00·29-s + 4.43·31-s + 12.1i·33-s − 0.860i·37-s − 10.2·39-s + ⋯ |
| L(s) = 1 | − 1.53i·3-s + 0.283i·7-s − 1.36·9-s − 1.37·11-s − 1.06i·13-s + 1.77i·17-s − 1.50·19-s + 0.436·21-s + 0.208i·23-s + 0.560i·27-s + 1.30·29-s + 0.795·31-s + 2.12i·33-s − 0.141i·37-s − 1.63·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4773867176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4773867176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 2.66iT - 3T^{2} \) |
| 7 | \( 1 - 0.750iT - 7T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 + 3.84iT - 13T^{2} \) |
| 17 | \( 1 - 7.32iT - 17T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 + 0.860iT - 37T^{2} \) |
| 41 | \( 1 + 9.60T + 41T^{2} \) |
| 43 | \( 1 - 6.57iT - 43T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 1.82iT - 53T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 - 10.1iT - 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 + 4.50iT - 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 11.9iT - 83T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
| 97 | \( 1 - 0.537iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.650607576335395904336884682846, −8.173047845706920277680137859305, −7.83271034856319669776465573535, −6.75269852597921181195200173059, −6.15092201982173902523794264344, −5.50265110416333624999926451386, −4.37600960312933760645326944091, −2.97622053773548575365803908795, −2.30530935114036861435566496101, −1.23740545325966807556889146921,
0.16392254755592012794825884119, 2.29706111638984976526487317633, 3.13929209735208587478758278026, 4.22807553007419660798841401363, 4.76230478717923276577726561587, 5.35961605984025563857658225194, 6.54912917220505411681958850703, 7.28590939708657248011497256014, 8.483400834540374691159165452717, 8.828430637792254262463581923571
