| L(s) = 1 | + (−1.25 − 0.642i)2-s + (−0.988 + 0.149i)3-s + (1.17 + 1.61i)4-s + (2.80 + 1.10i)5-s + (1.34 + 0.447i)6-s + (1.52 + 2.16i)7-s + (−0.438 − 2.79i)8-s + (0.955 − 0.294i)9-s + (−2.82 − 3.19i)10-s + (1.71 − 5.55i)11-s + (−1.40 − 1.42i)12-s + (0.363 − 0.0830i)13-s + (−0.534 − 3.70i)14-s + (−2.93 − 0.670i)15-s + (−1.24 + 3.80i)16-s + (3.94 + 5.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.890 − 0.454i)2-s + (−0.570 + 0.0860i)3-s + (0.587 + 0.809i)4-s + (1.25 + 0.492i)5-s + (0.547 + 0.182i)6-s + (0.576 + 0.816i)7-s + (−0.155 − 0.987i)8-s + (0.318 − 0.0982i)9-s + (−0.893 − 1.00i)10-s + (0.516 − 1.67i)11-s + (−0.404 − 0.411i)12-s + (0.100 − 0.0230i)13-s + (−0.142 − 0.989i)14-s + (−0.758 − 0.173i)15-s + (−0.310 + 0.950i)16-s + (0.955 + 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13080 + 0.0593133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13080 + 0.0593133i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 + 0.642i)T \) |
| 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 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−10.87795005307645970214488041811, −9.940923816145907990265939168661, −8.931384763783893982114912132856, −8.480620869015825188615459139297, −7.12190651453410732277458509936, −5.95931239020258900414462501364, −5.70805730785388806267102059878, −3.74947743757747248706235986357, −2.49728716613838976139178315732, −1.31023924865065261945865962489,
1.15286242924639541898645659260, 2.07816928964968379893844407582, 4.49167322930158693326915701232, 5.32657236990318658410260319674, 6.23613387322498964574867065557, 7.21874270198218228357002720448, 7.81469960421383284676755537364, 9.247514386995947157619989767952, 9.756816150559031863692516254516, 10.32105161362430515157433839461
