| L(s) = 1 | + (0.546 − 1.30i)2-s + (0.988 + 0.149i)3-s + (−1.40 − 1.42i)4-s + (2.80 − 1.10i)5-s + (0.734 − 1.20i)6-s + (−1.52 + 2.16i)7-s + (−2.62 + 1.04i)8-s + (0.955 + 0.294i)9-s + (0.0975 − 4.26i)10-s + (−1.71 − 5.55i)11-s + (−1.17 − 1.61i)12-s + (0.363 + 0.0830i)13-s + (1.98 + 3.17i)14-s + (2.93 − 0.670i)15-s + (−0.0670 + 3.99i)16-s + (3.94 − 5.77i)17-s + ⋯ |
| L(s) = 1 | + (0.386 − 0.922i)2-s + (0.570 + 0.0860i)3-s + (−0.701 − 0.713i)4-s + (1.25 − 0.492i)5-s + (0.300 − 0.493i)6-s + (−0.576 + 0.816i)7-s + (−0.928 + 0.371i)8-s + (0.318 + 0.0982i)9-s + (0.0308 − 1.34i)10-s + (−0.516 − 1.67i)11-s + (−0.338 − 0.467i)12-s + (0.100 + 0.0230i)13-s + (0.530 + 0.847i)14-s + (0.758 − 0.173i)15-s + (−0.0167 + 0.999i)16-s + (0.955 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35217 - 1.77940i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35217 - 1.77940i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.546 + 1.30i)T \) |
| 3 | \( 1 + (-0.988 - 0.149i)T \) |
| 7 | \( 1 + (1.52 - 2.16i)T \) |
| good | 5 | \( 1 + (-2.80 + 1.10i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (1.71 + 5.55i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.363 - 0.0830i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-3.94 + 5.77i)T + (-6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.74 + 3.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.103 - 0.152i)T + (-8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (0.320 - 0.154i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-4.31 - 7.48i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.592 - 7.91i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-5.81 - 4.63i)T + (9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (4.58 - 3.65i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (5.05 + 4.69i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.161 - 2.15i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (3.14 - 8.01i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (1.57 + 0.117i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (7.35 - 4.24i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.231 - 0.480i)T + (-44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (3.07 + 3.30i)T + (-5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-10.1 - 5.86i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.46 - 10.8i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (3.52 - 11.4i)T + (-73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 11.7iT - 97T^{2} \) |
| 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
−10.26296947256477523868466826022, −9.591040095404328963187342714176, −9.007314063962154837821903678780, −8.257920340870602818462588051541, −6.45583009440518784833315549546, −5.54245607551109081287092271064, −4.97636481394125867881875805332, −3.09995138466145252488830117744, −2.79194363535713884057937206808, −1.16302369152700113340485130604,
2.01621091015359467846952766313, 3.38954912700487352703030828203, 4.39933380446566066565138170634, 5.71210797794124713978371854733, 6.41666465145392088578241772016, 7.40016693146761426052642004867, 7.927535433270135498474653153854, 9.349525893369972383898074464180, 9.917724526884629270047374121348, 10.46023426500214778629052238392
