| L(s) = 1 | + (−1.13 − 2.48i)2-s + (−4.84 + 1.42i)3-s + (0.356 − 0.411i)4-s + (4.20 + 2.70i)5-s + (9.02 + 10.4i)6-s + (−3.26 − 22.7i)7-s + (−22.3 − 6.57i)8-s + (−1.29 + 0.830i)9-s + (1.94 − 13.5i)10-s + (−19.7 + 43.1i)11-s + (−1.14 + 2.50i)12-s + (3.63 − 25.3i)13-s + (−52.7 + 33.8i)14-s + (−24.2 − 7.10i)15-s + (8.44 + 58.7i)16-s + (56.7 + 65.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.401 − 0.878i)2-s + (−0.931 + 0.273i)3-s + (0.0445 − 0.0514i)4-s + (0.376 + 0.241i)5-s + (0.613 + 0.708i)6-s + (−0.176 − 1.22i)7-s + (−0.989 − 0.290i)8-s + (−0.0478 + 0.0307i)9-s + (0.0614 − 0.427i)10-s + (−0.540 + 1.18i)11-s + (−0.0274 + 0.0601i)12-s + (0.0776 − 0.539i)13-s + (−1.00 + 0.647i)14-s + (−0.416 − 0.122i)15-s + (0.131 + 0.917i)16-s + (0.809 + 0.934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0612224 + 0.0753821i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0612224 + 0.0753821i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.20 - 2.70i)T \) |
| 23 | \( 1 + (87.1 - 67.6i)T \) |
| good | 2 | \( 1 + (1.13 + 2.48i)T + (-5.23 + 6.04i)T^{2} \) |
| 3 | \( 1 + (4.84 - 1.42i)T + (22.7 - 14.5i)T^{2} \) |
| 7 | \( 1 + (3.26 + 22.7i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (19.7 - 43.1i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-3.63 + 25.3i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (-56.7 - 65.5i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (76.3 - 88.1i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (32.8 + 37.8i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (149. + 43.8i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (229. - 147. i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (-20.7 - 13.3i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (287. - 84.3i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 - 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (105. + 737. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-56.0 + 390. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-37.5 - 11.0i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (330. + 724. i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (-237. - 520. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (-128. + 148. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-42.7 + 297. i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (972. - 625. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-871. + 255. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (1.38e3 + 888. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
| 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
−12.98815512590282284759346813859, −12.14338529289985149005935831401, −10.98122573978632192925051475723, −10.19776431928860644644633726514, −10.00984876878327785951734665113, −7.983467510118603291249100209478, −6.53521211749335315041219369711, −5.44507939744148523173026819275, −3.72844039711461989523501347478, −1.77926098007103352911081250686,
0.06224546697333398272858682455, 2.75163514414393636770310066855, 5.42741216162228261164115944942, 5.95627744257267731536096155624, 7.02014625638075030019010475037, 8.557036061629525782373087075878, 9.108782134747149798455222901074, 10.88186382970625015870171293400, 11.88672645362731678727856666429, 12.50508135993430544720288128738
