L(s) = 1 | + (2.21 + 1.27i)2-s + (−2.59 + 1.5i)3-s + (−0.732 − 1.26i)4-s + (−11.1 − 0.184i)5-s − 7.66·6-s + (2.98 − 18.2i)7-s − 24.1i·8-s + (4.5 − 7.79i)9-s + (−24.5 − 14.6i)10-s + (0.664 + 1.15i)11-s + (3.80 + 2.19i)12-s − 46.0i·13-s + (29.9 − 36.6i)14-s + (29.3 − 16.2i)15-s + (25.0 − 43.4i)16-s + (−77.3 + 44.6i)17-s + ⋯ |
L(s) = 1 | + (0.782 + 0.451i)2-s + (−0.499 + 0.288i)3-s + (−0.0916 − 0.158i)4-s + (−0.999 − 0.0164i)5-s − 0.521·6-s + (0.161 − 0.986i)7-s − 1.06i·8-s + (0.166 − 0.288i)9-s + (−0.775 − 0.464i)10-s + (0.0182 + 0.0315i)11-s + (0.0916 + 0.0528i)12-s − 0.982i·13-s + (0.572 − 0.699i)14-s + (0.504 − 0.280i)15-s + (0.391 − 0.678i)16-s + (−1.10 + 0.636i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.662947 - 0.744028i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.662947 - 0.744028i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.59 - 1.5i)T \) |
| 5 | \( 1 + (11.1 + 0.184i)T \) |
| 7 | \( 1 + (-2.98 + 18.2i)T \) |
good | 2 | \( 1 + (-2.21 - 1.27i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.664 - 1.15i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 46.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (77.3 - 44.6i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (27.1 - 47.1i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (132. + 76.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (57.7 + 100. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-112. - 64.8i)T + (2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 428. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-487. - 281. i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-77.5 + 44.7i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-292. - 506. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-170. + 296. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-83.8 + 48.4i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 673.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (494. - 285. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-298. + 517. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 801. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-69.1 + 119. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.52e3iT - 9.12e5T^{2} \) |
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
−13.05260822018909097108697357177, −12.16459629498189449644562811189, −10.79780260888492139390981103937, −10.16218132515843977796278428053, −8.372191682219281380012204450385, −7.12635092220859813869756373956, −5.98769488871264229246549773995, −4.54195256060199429506193846858, −3.86089134200365741747231172423, −0.45880040025417417934873645555,
2.41992440369291059567539532549, 4.10002083936823364173381625827, 5.11422685051282230942026957792, 6.67018303673240949851036621067, 8.085567305185503009664391551232, 9.078072052562648282556522215985, 11.06358848598910289930674139992, 11.73132551188122044119332407294, 12.26969508261966665936023594726, 13.35453318113734390623349779169