| L(s) = 1 | + (0.404 − 1.35i)2-s + (0.909 − 0.415i)3-s + (−1.67 − 1.09i)4-s + (−0.108 + 0.0940i)5-s + (−0.195 − 1.40i)6-s + (−2.77 − 1.78i)7-s + (−2.16 + 1.82i)8-s + (0.654 − 0.755i)9-s + (0.0835 + 0.185i)10-s + (0.940 − 0.135i)11-s + (−1.97 − 0.302i)12-s + (−2.98 − 4.64i)13-s + (−3.53 + 3.03i)14-s + (−0.0596 + 0.130i)15-s + (1.59 + 3.66i)16-s + (−3.68 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (0.285 − 0.958i)2-s + (0.525 − 0.239i)3-s + (−0.836 − 0.548i)4-s + (−0.0485 + 0.0420i)5-s + (−0.0796 − 0.571i)6-s + (−1.04 − 0.673i)7-s + (−0.764 + 0.644i)8-s + (0.218 − 0.251i)9-s + (0.0264 + 0.0585i)10-s + (0.283 − 0.0407i)11-s + (−0.570 − 0.0871i)12-s + (−0.828 − 1.28i)13-s + (−0.945 + 0.811i)14-s + (−0.0153 + 0.0337i)15-s + (0.399 + 0.916i)16-s + (−0.894 + 0.262i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0572542 + 1.15825i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0572542 + 1.15825i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.404 + 1.35i)T \) |
| 3 | \( 1 + (-0.909 + 0.415i)T \) |
| 23 | \( 1 + (-3.55 + 3.21i)T \) |
| good | 5 | \( 1 + (0.108 - 0.0940i)T + (0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (2.77 + 1.78i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.940 + 0.135i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.98 + 4.64i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (3.68 - 1.08i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.383 + 1.30i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (2.93 + 10.0i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.26 - 4.95i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.129 - 0.112i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.0574 - 0.0662i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-7.05 + 3.22i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 9.35T + 47T^{2} \) |
| 53 | \( 1 + (0.253 - 0.395i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (4.42 + 6.87i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-8.48 - 3.87i)T + (39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-7.17 - 1.03i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.19 + 8.33i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (2.96 + 0.870i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (7.03 - 4.52i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.41 - 3.82i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (4.52 + 9.91i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.79 - 6.68i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.36047747868051699964759848989, −9.585596738271836192621103767078, −8.874386514945259389563429880293, −7.69001967325035540389620221374, −6.70900568128741378547714700240, −5.53645157488576840948879239130, −4.25272541010198433154239668195, −3.32784500119348598305299225705, −2.38894629510989325628090232322, −0.55587842771495001465235344673,
2.51983621278023123560260024763, 3.73827580690998055801012945993, 4.68751631648859370220503400735, 5.82575133236313212979718879848, 6.81099280797630271377599358909, 7.43119573288149016183793409270, 8.853591513681411690958066694750, 9.141644263087359176324284091155, 9.895617628419700932372954262900, 11.36863690453415726219358163943
