| L(s) = 1 | + (−0.716 + 1.21i)2-s + (−1.32 − 0.130i)3-s + (−0.972 − 1.74i)4-s + (−0.471 − 0.881i)5-s + (1.10 − 1.51i)6-s + (−0.600 − 3.01i)7-s + (2.82 + 0.0671i)8-s + (−1.20 − 0.240i)9-s + (1.41 + 0.0574i)10-s + (2.48 + 2.04i)11-s + (1.05 + 2.43i)12-s + (0.939 + 0.502i)13-s + (4.10 + 1.43i)14-s + (0.508 + 1.22i)15-s + (−2.10 + 3.39i)16-s + (1.16 − 2.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.506 + 0.862i)2-s + (−0.764 − 0.0752i)3-s + (−0.486 − 0.873i)4-s + (−0.210 − 0.394i)5-s + (0.452 − 0.620i)6-s + (−0.226 − 1.14i)7-s + (0.999 + 0.0237i)8-s + (−0.402 − 0.0801i)9-s + (0.446 + 0.0181i)10-s + (0.749 + 0.615i)11-s + (0.305 + 0.704i)12-s + (0.260 + 0.139i)13-s + (1.09 + 0.382i)14-s + (0.131 + 0.317i)15-s + (−0.527 + 0.849i)16-s + (0.282 − 0.680i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00157513 - 0.0532633i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00157513 - 0.0532633i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.716 - 1.21i)T \) |
| 5 | \( 1 + (0.471 + 0.881i)T \) |
| good | 3 | \( 1 + (1.32 + 0.130i)T + (2.94 + 0.585i)T^{2} \) |
| 7 | \( 1 + (0.600 + 3.01i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-2.48 - 2.04i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.939 - 0.502i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.16 + 2.80i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (5.85 - 1.77i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (2.47 - 1.65i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (2.57 + 3.14i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (7.47 - 7.47i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.989 - 3.26i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (-3.25 - 4.87i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-5.12 + 0.505i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (11.7 + 4.87i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-3.61 + 4.40i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (10.4 - 5.57i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (0.224 - 2.28i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (-0.108 + 1.10i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (12.1 - 2.40i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.11 - 5.60i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (6.75 - 2.79i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.34 - 4.43i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (-5.04 - 3.37i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (12.2 - 12.2i)T - 97iT^{2} \) |
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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
−10.85572252703687307924980172333, −10.09579166058619150792643124714, −9.217160845678389283073465943202, −8.341950863020627640120759712351, −7.31888154350398668679551370006, −6.65709632007525224847098830572, −5.80911747829316179829325609795, −4.74424280865999696605430338071, −3.86549088207498352611081797609, −1.39658231608443699015958726783,
0.03960091809381452886696941349, 2.03896936706372422518198599132, 3.21690856716442583874371090628, 4.30020236091498800875392918603, 5.72435890462268435983431435092, 6.31578604750475105947684465471, 7.72060202612221921082722850651, 8.744124800718925289212557492515, 9.152144948464973478003718674561, 10.44221476825000207215245625053
