| L(s) = 1 | + (−3.24 + 2.92i)3-s + (−2.16 − 3.75i)5-s + (−0.282 − 2.68i)7-s + (1.05 − 10.0i)9-s + (2.31 − 5.20i)11-s + (−4.07 − 19.1i)13-s + (18.0 + 5.84i)15-s + (−6.25 − 14.0i)17-s + (−1.74 − 0.371i)19-s + (8.77 + 7.89i)21-s + (−16.4 + 22.5i)23-s + (3.10 − 5.38i)25-s + (2.76 + 3.79i)27-s + (−46.6 + 15.1i)29-s + (11.5 + 28.7i)31-s + ⋯ |
| L(s) = 1 | + (−1.08 + 0.974i)3-s + (−0.433 − 0.750i)5-s + (−0.0403 − 0.384i)7-s + (0.117 − 1.11i)9-s + (0.210 − 0.473i)11-s + (−0.313 − 1.47i)13-s + (1.20 + 0.389i)15-s + (−0.367 − 0.826i)17-s + (−0.0919 − 0.0195i)19-s + (0.417 + 0.376i)21-s + (−0.713 + 0.981i)23-s + (0.124 − 0.215i)25-s + (0.102 + 0.140i)27-s + (−1.60 + 0.522i)29-s + (0.373 + 0.927i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.321300 - 0.377648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.321300 - 0.377648i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-11.5 - 28.7i)T \) |
| good | 3 | \( 1 + (3.24 - 2.92i)T + (0.940 - 8.95i)T^{2} \) |
| 5 | \( 1 + (2.16 + 3.75i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (0.282 + 2.68i)T + (-47.9 + 10.1i)T^{2} \) |
| 11 | \( 1 + (-2.31 + 5.20i)T + (-80.9 - 89.9i)T^{2} \) |
| 13 | \( 1 + (4.07 + 19.1i)T + (-154. + 68.7i)T^{2} \) |
| 17 | \( 1 + (6.25 + 14.0i)T + (-193. + 214. i)T^{2} \) |
| 19 | \( 1 + (1.74 + 0.371i)T + (329. + 146. i)T^{2} \) |
| 23 | \( 1 + (16.4 - 22.5i)T + (-163. - 503. i)T^{2} \) |
| 29 | \( 1 + (46.6 - 15.1i)T + (680. - 494. i)T^{2} \) |
| 37 | \( 1 + (14.2 + 8.20i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-1.03 + 1.14i)T + (-175. - 1.67e3i)T^{2} \) |
| 43 | \( 1 + (-7.28 + 34.2i)T + (-1.68e3 - 752. i)T^{2} \) |
| 47 | \( 1 + (-9.99 + 30.7i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-24.4 - 2.56i)T + (2.74e3 + 584. i)T^{2} \) |
| 59 | \( 1 + (64.2 + 71.3i)T + (-363. + 3.46e3i)T^{2} \) |
| 61 | \( 1 - 58.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (18.9 + 32.7i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (4.84 - 46.0i)T + (-4.93e3 - 1.04e3i)T^{2} \) |
| 73 | \( 1 + (-16.9 + 37.9i)T + (-3.56e3 - 3.96e3i)T^{2} \) |
| 79 | \( 1 + (-53.3 - 119. i)T + (-4.17e3 + 4.63e3i)T^{2} \) |
| 83 | \( 1 + (50.2 + 45.2i)T + (720. + 6.85e3i)T^{2} \) |
| 89 | \( 1 + (78.2 + 107. i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-134. + 97.7i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−12.64327539375641120962830913896, −11.73299615635859995359179616278, −10.81675826791865538128434264722, −9.972997728073286202140377268474, −8.773075313599588629573120565505, −7.42021569000776873278536241451, −5.73056142269696891765949375929, −4.93682803909919992136820144379, −3.67540460905958218839287198523, −0.38497907145311253100176410745,
2.01933133605118639919128446153, 4.26019026719031267474941997005, 5.98922895211683872337755509166, 6.75247869871892286763393751930, 7.68129273827173762598145857617, 9.263802349744845320647861542958, 10.72616481592342821079931282314, 11.58313584397708461586329337008, 12.20486734423147468001700049568, 13.19807525584561331136299573819
