| L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.730 − 1.57i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (−1.17 − 1.26i)6-s + (0.0245 + 0.234i)7-s + (0.587 − 0.809i)8-s + (−1.93 + 2.29i)9-s + (−0.669 + 0.743i)10-s + (−1.45 − 0.646i)11-s + (−1.51 − 0.841i)12-s + (−1.21 − 5.69i)13-s + (0.0957 + 0.214i)14-s + (1.41 + 0.995i)15-s + (0.309 − 0.951i)16-s + (−3.90 + 1.73i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (−0.421 − 0.906i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (−0.481 − 0.517i)6-s + (0.00929 + 0.0884i)7-s + (0.207 − 0.286i)8-s + (−0.644 + 0.764i)9-s + (−0.211 + 0.235i)10-s + (−0.437 − 0.194i)11-s + (−0.437 − 0.242i)12-s + (−0.335 − 1.58i)13-s + (0.0255 + 0.0574i)14-s + (0.366 + 0.256i)15-s + (0.0772 − 0.237i)16-s + (−0.947 + 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0169796 + 0.911372i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0169796 + 0.911372i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (0.730 + 1.57i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (5.48 - 0.974i)T \) |
| good | 7 | \( 1 + (-0.0245 - 0.234i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (1.45 + 0.646i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (1.21 + 5.69i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (3.90 - 1.73i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.572 - 0.121i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (7.46 + 5.42i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.549 + 1.69i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.834 - 0.481i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.53 - 4.98i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.201 + 0.945i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (7.81 + 2.53i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.193 + 1.84i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (2.88 - 2.59i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 5.54iT - 61T^{2} \) |
| 67 | \( 1 + (-2.84 - 4.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.9 - 1.46i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (6.83 - 15.3i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (5.16 + 11.5i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-10.2 + 11.3i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (14.8 - 10.7i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.67 + 4.11i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−10.00458516149616170535107843647, −8.428936398130069873643615956541, −7.895950635162061939041077135690, −6.97997615544533959309678193668, −6.06486489824046063998817685975, −5.41912937954412173633078568453, −4.30700345811146240022947852045, −3.01360140315932550220353496246, −2.09845526442290047747749793727, −0.33304156007845450034534173928,
2.16748747381296549475591018334, 3.65968204769139698204786054331, 4.31837740847134066432093284047, 5.07754608737209358030129527235, 6.01840862088036620629585848977, 6.95657565260128162534632236719, 7.83332927662737080880898474760, 9.056221345596577810715336019761, 9.549382462926649415383196361402, 10.67424333994688238379768811481
