| 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 | \( 1 + (11.9 - 7.41i)T + (43.2 - 86.8i)T^{2} \) |
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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
−10.75842951773069450796995779321, −10.06442777813897935072338938592, −8.234185397133862542327476242792, −8.003467219154326499296804375819, −6.84004682575214064063706723514, −6.24206120774273348547170017790, −5.50449278523206795087668822916, −4.06937002219015140880665380568, −2.66926269069168582092677717318, −1.27393562002122320052027789115,
1.51696590364659706335515634667, 3.20122674658219076365928163748, 4.32618099048248549993534115825, 4.99146005774398968048500553165, 5.73725075100954448742885784604, 6.89045756137160276911853076569, 8.737402897694788458099091665539, 9.043962270848695335498456396032, 10.00874394422427901727485731142, 10.98536683767710477724945658576
