| L(s) = 1 | + (−1.33 + 0.466i)2-s + (−0.988 + 0.149i)3-s + (1.56 − 1.24i)4-s + (2.58 + 1.01i)5-s + (1.25 − 0.660i)6-s + (1.51 + 2.16i)7-s + (−1.50 + 2.39i)8-s + (0.955 − 0.294i)9-s + (−3.92 − 0.148i)10-s + (−1.46 + 4.74i)11-s + (−1.36 + 1.46i)12-s + (1.62 − 0.371i)13-s + (−3.03 − 2.19i)14-s + (−2.70 − 0.617i)15-s + (0.898 − 3.89i)16-s + (−1.65 − 2.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.944 + 0.329i)2-s + (−0.570 + 0.0860i)3-s + (0.782 − 0.622i)4-s + (1.15 + 0.453i)5-s + (0.510 − 0.269i)6-s + (0.572 + 0.819i)7-s + (−0.533 + 0.845i)8-s + (0.318 − 0.0982i)9-s + (−1.24 − 0.0471i)10-s + (−0.441 + 1.43i)11-s + (−0.393 + 0.422i)12-s + (0.451 − 0.103i)13-s + (−0.810 − 0.585i)14-s + (−0.698 − 0.159i)15-s + (0.224 − 0.974i)16-s + (−0.401 − 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.633510 + 0.728029i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.633510 + 0.728029i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.466i)T \) |
| 3 | \( 1 + (0.988 - 0.149i)T \) |
| 7 | \( 1 + (-1.51 - 2.16i)T \) |
| good | 5 | \( 1 + (-2.58 - 1.01i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (1.46 - 4.74i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-1.62 + 0.371i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (1.65 + 2.42i)T + (-6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-0.522 - 0.904i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.413 - 0.606i)T + (-8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-3.85 - 1.85i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.702 + 1.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.131 + 1.74i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (4.25 - 3.38i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (7.54 + 6.01i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (1.55 - 1.44i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (0.844 - 11.2i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-2.28 - 5.83i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-14.9 + 1.11i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (0.627 + 0.362i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.13 + 2.34i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (7.13 - 7.69i)T + (-5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (-13.9 + 8.03i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.263 + 1.15i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-3.12 - 10.1i)T + (-73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−10.63005770826928293154400733927, −10.01788845896440667840117381144, −9.356511112766517127390983872690, −8.384784457424124299978139830811, −7.28133908161219554682382837287, −6.48257986840456489826097538235, −5.61475237779506527086938521267, −4.86713922208741295868960829266, −2.56100997496850809623527782336, −1.68305253738961842149096190485,
0.817815217136928042029318439567, 1.93865241088478381361524138698, 3.53044460357293044163565535276, 5.02133472055266487072489015468, 6.08667352815549435382744813467, 6.79622244677115784085886713890, 8.156487025130022259668759734126, 8.601926258216853077206887477664, 9.800095471387343678978139543104, 10.41655489026513155750233365710
