| L(s) = 1 | + (1.84 + 4.85i)3-s + (1.01 − 1.75i)5-s + (−20.1 + 17.9i)9-s + (25.2 − 14.5i)11-s − 25.7i·13-s + (10.4 + 1.67i)15-s + (−1.39 − 2.42i)17-s + (92.8 + 53.6i)19-s + (70.3 + 40.6i)23-s + (60.4 + 104. i)25-s + (−124. − 64.6i)27-s + 170. i·29-s + (22.4 − 12.9i)31-s + (117. + 95.7i)33-s + (−90.6 + 156. i)37-s + ⋯ |
| L(s) = 1 | + (0.355 + 0.934i)3-s + (0.0907 − 0.157i)5-s + (−0.746 + 0.665i)9-s + (0.693 − 0.400i)11-s − 0.548i·13-s + (0.179 + 0.0288i)15-s + (−0.0199 − 0.0345i)17-s + (1.12 + 0.647i)19-s + (0.637 + 0.368i)23-s + (0.483 + 0.837i)25-s + (−0.887 − 0.460i)27-s + 1.09i·29-s + (0.130 − 0.0750i)31-s + (0.620 + 0.505i)33-s + (−0.402 + 0.697i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00875 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00875 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.227057978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.227057978\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.84 - 4.85i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-1.01 + 1.75i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-25.2 + 14.5i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 25.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (1.39 + 2.42i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-92.8 - 53.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-70.3 - 40.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 170. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-22.4 + 12.9i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (90.6 - 156. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 257.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 239.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (186. - 323. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-489. + 282. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-189. - 328. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (194. + 112. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (167. + 290. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.02e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-505. + 292. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-497. + 861. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (738. - 1.27e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.22e3iT - 9.12e5T^{2} \) |
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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
−10.43133368796373603662554724284, −9.567194269650770592247941739608, −8.916413681065644019344871961008, −8.063596871216955295782871550197, −6.98603537689426273535769456526, −5.65591073397402972253288334774, −4.99431502569886080038892520227, −3.70945558021346375253685069908, −2.99559338340090110107705544189, −1.25980433591629567282240354064,
0.69782598317606412278591545198, 1.96600795216194380406153453295, 3.04862643002557592750714504980, 4.32588791798401565609860763441, 5.64255691794579392330647108207, 6.75116746481267919556535007646, 7.16937144598307485733455517571, 8.325255382272602399080562991972, 9.089286423784348579234160814810, 9.903174723116107799558302485989
