| L(s) = 1 | + (−1.33 − 0.477i)2-s + (0.934 − 1.45i)3-s + (1.54 + 1.27i)4-s + (1.57 − 1.79i)5-s + (−1.94 + 1.49i)6-s + (−0.379 − 2.88i)7-s + (−1.44 − 2.43i)8-s + (−1.25 − 2.72i)9-s + (−2.95 + 1.63i)10-s + (4.15 + 2.05i)11-s + (3.29 − 1.06i)12-s + (−4.81 − 1.63i)13-s + (−0.872 + 4.01i)14-s + (−1.14 − 3.97i)15-s + (0.765 + 3.92i)16-s + (0.291 + 0.291i)17-s + ⋯ |
| L(s) = 1 | + (−0.941 − 0.337i)2-s + (0.539 − 0.841i)3-s + (0.771 + 0.635i)4-s + (0.703 − 0.802i)5-s + (−0.792 + 0.610i)6-s + (−0.143 − 1.08i)7-s + (−0.511 − 0.859i)8-s + (−0.417 − 0.908i)9-s + (−0.933 + 0.517i)10-s + (1.25 + 0.618i)11-s + (0.951 − 0.306i)12-s + (−1.33 − 0.453i)13-s + (−0.233 + 1.07i)14-s + (−0.295 − 1.02i)15-s + (0.191 + 0.981i)16-s + (0.0706 + 0.0706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.431679 - 1.12848i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.431679 - 1.12848i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 + 0.477i)T \) |
| 3 | \( 1 + (-0.934 + 1.45i)T \) |
| good | 5 | \( 1 + (-1.57 + 1.79i)T + (-0.652 - 4.95i)T^{2} \) |
| 7 | \( 1 + (0.379 + 2.88i)T + (-6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-4.15 - 2.05i)T + (6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (4.81 + 1.63i)T + (10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-0.291 - 0.291i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.13 + 5.69i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (1.09 - 8.30i)T + (-22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (-5.91 - 0.387i)T + (28.7 + 3.78i)T^{2} \) |
| 31 | \( 1 + (1.09 - 1.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.87 - 1.56i)T + (34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-8.57 - 1.12i)T + (39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-0.566 - 0.279i)T + (26.1 + 34.1i)T^{2} \) |
| 47 | \( 1 + (12.3 + 3.30i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (5.24 + 7.84i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-5.10 - 4.48i)T + (7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (8.57 + 0.562i)T + (60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (8.63 - 4.26i)T + (40.7 - 53.1i)T^{2} \) |
| 71 | \( 1 + (-1.92 + 4.65i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (2.14 + 5.18i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (6.60 + 1.77i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-7.35 - 8.38i)T + (-10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-3.09 + 7.46i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-12.4 + 7.16i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.931014697766859605057592697266, −9.539191149523002398089727928386, −8.811471645217195923070572302825, −7.63853851625367022693515490201, −7.15114007001228864567739191274, −6.23795975132553109209565264053, −4.59152096505884339701415205263, −3.20473489335444856119510307182, −1.87291534683510018993411434694, −0.885461843574364626738017889580,
2.16001270635721369943563794322, 2.87456908818606540938560086859, 4.55601489607437446976234207784, 6.00329210034058137957767768218, 6.35979112755041547818047515960, 7.75011637425404500879844634162, 8.668778321461185940123243596486, 9.361610507251153689822055074650, 9.936093257296701308381812347613, 10.67360427750457920890916924339
