| L(s) = 1 | + (1.38 − 0.288i)2-s + (−0.988 − 0.149i)3-s + (1.83 − 0.798i)4-s + (−2.36 + 0.929i)5-s + (−1.41 + 0.0788i)6-s + (−2.64 − 0.0802i)7-s + (2.30 − 1.63i)8-s + (0.955 + 0.294i)9-s + (−3.01 + 1.97i)10-s + (−1.59 − 5.16i)11-s + (−1.93 + 0.516i)12-s + (−5.52 − 1.26i)13-s + (−3.68 + 0.651i)14-s + (2.48 − 0.566i)15-s + (2.72 − 2.92i)16-s + (−1.00 + 1.47i)17-s + ⋯ |
| L(s) = 1 | + (0.978 − 0.203i)2-s + (−0.570 − 0.0860i)3-s + (0.916 − 0.399i)4-s + (−1.05 + 0.415i)5-s + (−0.576 + 0.0321i)6-s + (−0.999 − 0.0303i)7-s + (0.816 − 0.577i)8-s + (0.318 + 0.0982i)9-s + (−0.952 + 0.623i)10-s + (−0.480 − 1.55i)11-s + (−0.557 + 0.149i)12-s + (−1.53 − 0.349i)13-s + (−0.984 + 0.174i)14-s + (0.640 − 0.146i)15-s + (0.681 − 0.732i)16-s + (−0.243 + 0.356i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.215696 - 0.792010i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.215696 - 0.792010i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 + 0.288i)T \) |
| 3 | \( 1 + (0.988 + 0.149i)T \) |
| 7 | \( 1 + (2.64 + 0.0802i)T \) |
| good | 5 | \( 1 + (2.36 - 0.929i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (1.59 + 5.16i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (5.52 + 1.26i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (1.00 - 1.47i)T + (-6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-3.25 + 5.63i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.62 + 2.37i)T + (-8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (2.37 - 1.14i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-5.27 - 9.13i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.00446 - 0.0595i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (3.22 + 2.57i)T + (9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (3.96 - 3.16i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (6.84 + 6.34i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.512 - 6.83i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-1.96 + 4.99i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-8.01 - 0.600i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-5.15 + 2.97i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.44 - 7.16i)T + (-44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-2.64 - 2.85i)T + (-5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (2.78 + 1.60i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.65 + 16.0i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (1.78 - 5.77i)T + (-73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 4.18iT - 97T^{2} \) |
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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
−10.61180019056309061253572413286, −9.870631730553427111432681994971, −8.374146715436600942634032119248, −7.22290045570252877795651139988, −6.71779277158922098983466679956, −5.60389521958547072485461946365, −4.74115201428274845456108703977, −3.43582725646133089940670685005, −2.82273277107312995417146725115, −0.33263537207050555515601557074,
2.29922817376804043664963732575, 3.73887288760651190657620273115, 4.55190858741962261180519845455, 5.33486177312389782258020396979, 6.53049320702835606422823973950, 7.41073118957411343514665588152, 7.88358717959284285627941205039, 9.751291128733262807412740915491, 10.02939240012562287870661473687, 11.65006873585785721482776389818
