| L(s) = 1 | + (0.623 + 0.781i)2-s + (−2.78 + 1.34i)3-s + (−0.222 + 0.974i)4-s + (−0.309 + 0.148i)5-s + (−2.78 − 1.34i)6-s + (−2.03 + 1.69i)7-s + (−0.900 + 0.433i)8-s + (4.10 − 5.14i)9-s + (−0.309 − 0.148i)10-s + (2.82 + 3.54i)11-s + (−0.688 − 3.01i)12-s + (1.37 + 1.71i)13-s + (−2.59 − 0.536i)14-s + (0.662 − 0.830i)15-s + (−0.900 − 0.433i)16-s + (−0.0797 − 0.349i)17-s + ⋯ |
| L(s) = 1 | + (0.440 + 0.552i)2-s + (−1.60 + 0.775i)3-s + (−0.111 + 0.487i)4-s + (−0.138 + 0.0665i)5-s + (−1.13 − 0.548i)6-s + (−0.769 + 0.639i)7-s + (−0.318 + 0.153i)8-s + (1.36 − 1.71i)9-s + (−0.0977 − 0.0470i)10-s + (0.852 + 1.06i)11-s + (−0.198 − 0.871i)12-s + (0.380 + 0.476i)13-s + (−0.692 − 0.143i)14-s + (0.170 − 0.214i)15-s + (−0.225 − 0.108i)16-s + (−0.0193 − 0.0847i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.245771 + 0.643964i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.245771 + 0.643964i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 7 | \( 1 + (2.03 - 1.69i)T \) |
| good | 3 | \( 1 + (2.78 - 1.34i)T + (1.87 - 2.34i)T^{2} \) |
| 5 | \( 1 + (0.309 - 0.148i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-2.82 - 3.54i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.37 - 1.71i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.0797 + 0.349i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + (-1.31 + 5.75i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (0.436 + 1.91i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + (-2.55 - 11.1i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-9.04 + 4.35i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (5.90 + 2.84i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.545 - 0.684i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-1.77 + 7.77i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-2.59 - 1.25i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-1.68 - 7.36i)T + (-54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 + (-0.0251 + 0.109i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (0.453 - 0.568i)T + (-16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + 3.61T + 79T^{2} \) |
| 83 | \( 1 + (-8.63 + 10.8i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-2.93 + 3.68i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−14.69797304763118849653829960603, −13.04540309471921163661559446564, −12.03895873692243215026943313237, −11.49825349921864783192291079272, −10.02359130091924717044361806313, −9.169015246563870719972117875463, −7.01354689518618641543319189919, −6.18895376154031469788237277662, −5.08361932182992289129464361965, −3.89219699608964899627256023990,
0.930845905287256265163621887188, 3.74460418474842040313896121743, 5.51120943782617457149804506490, 6.30249114061468922688265196673, 7.50842760317665969697869028924, 9.541395821276494802185180547993, 10.87460453450517048868735412606, 11.39608798945268501437783234688, 12.42402255094691178388924440779, 13.21360419896441718670389163146
