| L(s) = 1 | + (−0.163 + 1.03i)2-s + (2.50 + 1.27i)3-s + (0.863 + 0.280i)4-s + (−1.84 + 1.27i)5-s + (−1.72 + 2.38i)6-s + (−0.249 − 0.489i)7-s + (−1.37 + 2.70i)8-s + (2.89 + 3.98i)9-s + (−1.01 − 2.10i)10-s + (1.80 + 1.80i)12-s + (−3.20 − 0.507i)13-s + (0.546 − 0.177i)14-s + (−6.24 + 0.834i)15-s + (−1.09 − 0.798i)16-s + (5.59 − 0.885i)17-s + (−4.59 + 2.33i)18-s + ⋯ |
| L(s) = 1 | + (−0.115 + 0.729i)2-s + (1.44 + 0.738i)3-s + (0.431 + 0.140i)4-s + (−0.822 + 0.568i)5-s + (−0.706 + 0.971i)6-s + (−0.0943 − 0.185i)7-s + (−0.487 + 0.957i)8-s + (0.966 + 1.32i)9-s + (−0.319 − 0.666i)10-s + (0.522 + 0.522i)12-s + (−0.888 − 0.140i)13-s + (0.146 − 0.0474i)14-s + (−1.61 + 0.215i)15-s + (−0.274 − 0.199i)16-s + (1.35 − 0.214i)17-s + (−1.08 + 0.551i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.787879 + 1.99863i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.787879 + 1.99863i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.84 - 1.27i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.163 - 1.03i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (-2.50 - 1.27i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (0.249 + 0.489i)T + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (3.20 + 0.507i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-5.59 + 0.885i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (0.480 + 1.48i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.803 - 0.803i)T - 23iT^{2} \) |
| 29 | \( 1 + (1.31 - 4.04i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.33 + 0.968i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.852 + 0.434i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-8.50 + 2.76i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-2.55 - 2.55i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.84 + 3.62i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (1.10 - 6.95i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-6.43 - 2.09i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.40 + 7.44i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.62 + 2.62i)T + 67iT^{2} \) |
| 71 | \( 1 + (5.54 + 4.02i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.98 - 4.58i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (3.37 - 2.44i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.77 - 11.1i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + 3.64iT - 89T^{2} \) |
| 97 | \( 1 + (16.4 + 2.60i)T + (92.2 + 29.9i)T^{2} \) |
| 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
−10.76745769671324152500323854999, −9.973833165143805640804167424030, −9.056837474589003453469919918976, −8.133130528555952374675454290805, −7.58047888620491481500674294093, −6.97355404663982866346215042771, −5.51414397625188318261547140238, −4.21477366231885826583732054214, −3.21284751197985644094190334067, −2.50689059951633759507079440283,
1.11390250276006144624267806789, 2.33788454957049229831682066603, 3.21319562475697098495095068674, 4.18424890995917959486169647968, 5.85880611256045598886069271325, 7.21326164804634629152960222408, 7.68566937619950911212995243840, 8.564540318227383272430813757080, 9.453813669251335219242740974561, 10.13011804756992438495982293495
