| L(s) = 1 | + (−1.26 − 0.340i)2-s + (−1.59 + 0.664i)3-s + (−0.236 − 0.136i)4-s + (2.23 − 0.155i)5-s + (2.25 − 0.299i)6-s + (1.25 + 2.32i)7-s + (2.11 + 2.11i)8-s + (2.11 − 2.12i)9-s + (−2.88 − 0.560i)10-s + (3.38 + 1.95i)11-s + (0.468 + 0.0611i)12-s + (−1.56 + 1.56i)13-s + (−0.807 − 3.38i)14-s + (−3.46 + 1.73i)15-s + (−1.69 − 2.92i)16-s + (−0.693 − 2.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.897 − 0.240i)2-s + (−0.923 + 0.383i)3-s + (−0.118 − 0.0681i)4-s + (0.997 − 0.0697i)5-s + (0.921 − 0.122i)6-s + (0.476 + 0.879i)7-s + (0.746 + 0.746i)8-s + (0.705 − 0.708i)9-s + (−0.912 − 0.177i)10-s + (1.01 + 0.588i)11-s + (0.135 + 0.0176i)12-s + (−0.434 + 0.434i)13-s + (−0.215 − 0.903i)14-s + (−0.894 + 0.447i)15-s + (−0.422 − 0.731i)16-s + (−0.168 − 0.627i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.572052 + 0.116772i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.572052 + 0.116772i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.59 - 0.664i)T \) |
| 5 | \( 1 + (-2.23 + 0.155i)T \) |
| 7 | \( 1 + (-1.25 - 2.32i)T \) |
| good | 2 | \( 1 + (1.26 + 0.340i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (-3.38 - 1.95i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.56 - 1.56i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.693 + 2.58i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 0.930i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.638 - 2.38i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 0.513T + 29T^{2} \) |
| 31 | \( 1 + (4.29 - 7.43i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.77 + 6.60i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 0.308iT - 41T^{2} \) |
| 43 | \( 1 + (-7.60 + 7.60i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.10 + 1.36i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.85 + 0.498i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.259 - 0.448i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.55 + 4.42i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.74 - 2.34i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 + (0.749 + 2.79i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (4.37 - 2.52i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (9.16 + 9.16i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.67 - 9.82i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.81 + 6.81i)T + 97iT^{2} \) |
| 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
−13.96956584825560252787638841650, −12.44936103572861679069922631887, −11.53331221984595996746831773782, −10.50805149112917444510601830433, −9.347623361032583298565125613893, −9.124047381503724995983117002220, −7.11672428074250424854903028499, −5.65823479227484294269606180645, −4.72969088619154278259603818389, −1.73904343822875846850639853300,
1.24324151723839757288615306673, 4.36729678002318017081788846191, 5.95298201019401092014987176496, 7.04958905861857775820358890950, 8.145833308543413029373144095922, 9.540267385189872841488655235290, 10.37958817601971674607072615681, 11.32482102268712550650756418832, 12.79677072743249089405635632074, 13.54885294685119897336684567112
