| L(s) = 1 | + (0.739 − 0.673i)2-s + (−0.641 + 2.25i)3-s + (0.0922 − 0.995i)4-s + (0.981 − 1.29i)5-s + (1.04 + 2.09i)6-s + (1.36 + 0.846i)7-s + (−0.602 − 0.798i)8-s + (−2.11 − 1.31i)9-s + (−0.150 − 1.62i)10-s + (0.139 − 1.50i)11-s + (2.18 + 0.846i)12-s + (3.71 + 4.91i)13-s + (1.58 − 0.295i)14-s + (2.29 + 3.04i)15-s + (−0.982 − 0.183i)16-s + (3.55 − 2.08i)17-s + ⋯ |
| L(s) = 1 | + (0.522 − 0.476i)2-s + (−0.370 + 1.30i)3-s + (0.0461 − 0.497i)4-s + (0.438 − 0.580i)5-s + (0.426 + 0.856i)6-s + (0.516 + 0.320i)7-s + (−0.213 − 0.282i)8-s + (−0.706 − 0.437i)9-s + (−0.0475 − 0.512i)10-s + (0.0421 − 0.454i)11-s + (0.630 + 0.244i)12-s + (1.02 + 1.36i)13-s + (0.422 − 0.0789i)14-s + (0.593 + 0.786i)15-s + (−0.245 − 0.0459i)16-s + (0.862 − 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.97794 + 0.342667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.97794 + 0.342667i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.739 + 0.673i)T \) |
| 17 | \( 1 + (-3.55 + 2.08i)T \) |
| good | 3 | \( 1 + (0.641 - 2.25i)T + (-2.55 - 1.57i)T^{2} \) |
| 5 | \( 1 + (-0.981 + 1.29i)T + (-1.36 - 4.80i)T^{2} \) |
| 7 | \( 1 + (-1.36 - 0.846i)T + (3.12 + 6.26i)T^{2} \) |
| 11 | \( 1 + (-0.139 + 1.50i)T + (-10.8 - 2.02i)T^{2} \) |
| 13 | \( 1 + (-3.71 - 4.91i)T + (-3.55 + 12.5i)T^{2} \) |
| 19 | \( 1 + (4.93 + 4.49i)T + (1.75 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-4.80 - 2.97i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (-0.0714 + 0.771i)T + (-28.5 - 5.32i)T^{2} \) |
| 31 | \( 1 + (-5.33 - 7.06i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (2.44 - 0.947i)T + (27.3 - 24.9i)T^{2} \) |
| 41 | \( 1 + (-2.65 + 9.31i)T + (-34.8 - 21.5i)T^{2} \) |
| 43 | \( 1 + (1.83 - 0.343i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + (7.07 - 4.37i)T + (20.9 - 42.0i)T^{2} \) |
| 53 | \( 1 + (-11.1 - 6.93i)T + (23.6 + 47.4i)T^{2} \) |
| 59 | \( 1 + (0.799 - 1.60i)T + (-35.5 - 47.0i)T^{2} \) |
| 61 | \( 1 + (5.28 + 10.6i)T + (-36.7 + 48.6i)T^{2} \) |
| 67 | \( 1 + (-2.44 - 2.23i)T + (6.18 + 66.7i)T^{2} \) |
| 71 | \( 1 + (2.04 + 1.26i)T + (31.6 + 63.5i)T^{2} \) |
| 73 | \( 1 + (-0.316 - 0.0591i)T + (68.0 + 26.3i)T^{2} \) |
| 79 | \( 1 + (10.5 + 9.59i)T + (7.28 + 78.6i)T^{2} \) |
| 83 | \( 1 + (3.24 + 11.3i)T + (-70.5 + 43.6i)T^{2} \) |
| 89 | \( 1 + (1.89 - 2.51i)T + (-24.3 - 85.6i)T^{2} \) |
| 97 | \( 1 + (11.9 + 7.41i)T + (43.2 + 86.8i)T^{2} \) |
| 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
−10.98536683767710477724945658576, −10.00874394422427901727485731142, −9.043962270848695335498456396032, −8.737402897694788458099091665539, −6.89045756137160276911853076569, −5.73725075100954448742885784604, −4.99146005774398968048500553165, −4.32618099048248549993534115825, −3.20122674658219076365928163748, −1.51696590364659706335515634667,
1.27393562002122320052027789115, 2.66926269069168582092677717318, 4.06937002219015140880665380568, 5.50449278523206795087668822916, 6.24206120774273348547170017790, 6.84004682575214064063706723514, 8.003467219154326499296804375819, 8.234185397133862542327476242792, 10.06442777813897935072338938592, 10.75842951773069450796995779321
