| L(s) = 1 | + (0.142 − 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.959 − 0.281i)6-s + (−2.31 − 1.48i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.841 + 0.540i)10-s + (0.0681 + 0.474i)11-s + (−0.142 − 0.989i)12-s + (−3.48 + 2.23i)13-s + (−1.80 + 2.07i)14-s + (0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (−2.59 + 0.762i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (0.391 − 0.115i)6-s + (−0.874 − 0.561i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.266 + 0.170i)10-s + (0.0205 + 0.142i)11-s + (−0.0410 − 0.285i)12-s + (−0.966 + 0.620i)13-s + (−0.481 + 0.555i)14-s + (0.107 − 0.234i)15-s + (0.210 + 0.135i)16-s + (−0.630 + 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0187729 + 0.0492976i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0187729 + 0.0492976i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (3.77 - 2.95i)T \) |
| good | 7 | \( 1 + (2.31 + 1.48i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.0681 - 0.474i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.48 - 2.23i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (2.59 - 0.762i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.983 + 0.288i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (3.20 - 0.942i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.829 + 1.81i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (0.403 - 0.465i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (1.49 + 1.72i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.25 - 2.73i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 + (9.58 + 6.16i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (8.64 - 5.55i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.15 + 9.09i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.109 + 0.764i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.16 - 8.09i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-8.90 - 2.61i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-6.15 + 3.95i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.46 + 2.85i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.300 - 0.657i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (3.39 + 3.91i)T + (-13.8 + 96.0i)T^{2} \) |
| 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
−10.77400266651737297896840061978, −9.737536346830582292980708136202, −9.516557614617521211445011593154, −8.415895061984356623484978969569, −7.39252886128047508556567894104, −6.33025953833229929457656940186, −5.00353059369115348212604904605, −4.17526034899905161148077105747, −3.35619231848105860329455555246, −2.03171735499781664560687361863,
0.02422999134927433155676930294, 2.41643712758886225409415223934, 3.41929603786419917247710362179, 4.73722332566060900465994784188, 5.93454259252434861508228170295, 6.58102453908523779961058038616, 7.47079796011108438945414560419, 8.221377770460488713514859243267, 9.158409250300506170507403203337, 9.910785882125358362715283333795
