| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (0.173 − 0.984i)6-s + (2.43 − 4.22i)7-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.939 − 0.342i)10-s + (−2.26 − 3.92i)11-s + (−0.5 + 0.866i)12-s + (−0.205 + 1.16i)13-s + (−3.73 + 3.13i)14-s + (−0.766 − 0.642i)15-s + (0.173 + 0.984i)16-s + (−1.32 − 0.482i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (−0.342 + 0.287i)5-s + (0.0708 − 0.402i)6-s + (0.922 − 1.59i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (0.297 − 0.108i)10-s + (−0.683 − 1.18i)11-s + (−0.144 + 0.249i)12-s + (−0.0570 + 0.323i)13-s + (−0.998 + 0.838i)14-s + (−0.197 − 0.165i)15-s + (0.0434 + 0.246i)16-s + (−0.321 − 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0540 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0540 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.613434 - 0.581099i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.613434 - 0.581099i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (4.21 + 1.10i)T \) |
| good | 7 | \( 1 + (-2.43 + 4.22i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.26 + 3.92i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.205 - 1.16i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.32 + 0.482i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (2.73 + 2.29i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.0393 - 0.0143i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.95 + 8.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.35T + 37T^{2} \) |
| 41 | \( 1 + (0.928 + 5.26i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.748 - 0.628i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.70 + 0.620i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-8.99 - 7.54i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.15 - 0.419i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (5.20 + 4.37i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-5.13 + 1.86i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (7.63 - 6.40i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.52 - 14.3i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.25 + 7.11i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.26 + 7.39i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.295 + 1.67i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (6.57 + 2.39i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.64183040620461118936028126963, −9.886151302023706070688691738819, −8.619038492287897808732355362797, −8.032620559243871304482903494083, −7.23764307497103412151954559524, −6.07111108379379111493974199585, −4.52685340048696072569636856266, −3.87553885475496998705710391757, −2.46902093533486003756647698944, −0.58270084889359487874146082038,
1.73323821703886872615014676849, 2.60750199481775273155705840305, 4.64753316825832257944332092087, 5.51625255496682729149524508781, 6.54046166600648565638032976403, 7.71128930855301686264288821660, 8.252733669483017387727808169823, 8.914917094465985543471735677729, 9.942750063058160540498039632982, 10.96042934623248639095231197451
