| L(s) = 1 | + (0.856 − 0.112i)2-s + (0.571 + 1.15i)3-s + (−1.21 + 0.324i)4-s + (−0.751 + 0.659i)5-s + (0.620 + 0.929i)6-s + (−0.581 + 2.58i)7-s + (−2.59 + 1.07i)8-s + (0.808 − 1.05i)9-s + (−0.569 + 0.649i)10-s + (−4.57 + 0.299i)11-s + (−1.06 − 1.21i)12-s + (−2.22 − 2.22i)13-s + (−0.207 + 2.27i)14-s + (−1.19 − 0.494i)15-s + (0.0651 − 0.0376i)16-s + (−3.61 − 1.97i)17-s + ⋯ |
| L(s) = 1 | + (0.605 − 0.0797i)2-s + (0.330 + 0.669i)3-s + (−0.605 + 0.162i)4-s + (−0.336 + 0.294i)5-s + (0.253 + 0.379i)6-s + (−0.219 + 0.975i)7-s + (−0.918 + 0.380i)8-s + (0.269 − 0.351i)9-s + (−0.180 + 0.205i)10-s + (−1.37 + 0.0903i)11-s + (−0.308 − 0.351i)12-s + (−0.617 − 0.617i)13-s + (−0.0554 + 0.608i)14-s + (−0.308 − 0.127i)15-s + (0.0162 − 0.00940i)16-s + (−0.877 − 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.102724 + 0.826738i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.102724 + 0.826738i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.751 - 0.659i)T \) |
| 7 | \( 1 + (0.581 - 2.58i)T \) |
| 17 | \( 1 + (3.61 + 1.97i)T \) |
| good | 2 | \( 1 + (-0.856 + 0.112i)T + (1.93 - 0.517i)T^{2} \) |
| 3 | \( 1 + (-0.571 - 1.15i)T + (-1.82 + 2.38i)T^{2} \) |
| 11 | \( 1 + (4.57 - 0.299i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (2.22 + 2.22i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.376 - 2.86i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-4.95 - 2.44i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (1.01 + 5.09i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (5.66 - 2.79i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.776 - 11.8i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (1.80 - 9.07i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.51 - 6.08i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (2.43 - 9.08i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (6.04 + 7.88i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (-9.54 - 1.25i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.987 - 2.90i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (3.91 + 2.25i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.75 + 2.50i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-9.76 - 3.31i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-2.47 + 5.01i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-1.89 + 4.57i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-0.0181 - 0.00486i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (2.95 - 0.587i)T + (89.6 - 37.1i)T^{2} \) |
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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
−11.16286330631852896965491144483, −9.962133563585021080392440117224, −9.485155453873214981851435349774, −8.510432469467395509100309406713, −7.73714825966442803147848687548, −6.36318336932745517640918068668, −5.18484116912930157217031677542, −4.66958602216116434825125390601, −3.30602230688620189137208636743, −2.76158492227428496023824947552,
0.35530130599937115189284187977, 2.26728549013144881539490428100, 3.70598878568650396611306383209, 4.65994992709580250124967765659, 5.39918977835755659354358080741, 6.94999513692231540148856540445, 7.34888981763535895536511343827, 8.508019458642935585647634787934, 9.232394723042902222282683430729, 10.43807104001507314093628402066
