| L(s) = 1 | + (−0.756 + 1.65i)3-s + (−2.72 + 3.14i)5-s + (1.70 − 1.09i)7-s + (−0.204 − 0.236i)9-s + (0.491 − 3.41i)11-s + (4.95 + 3.18i)13-s + (−3.14 − 6.89i)15-s + (−2.45 − 0.722i)17-s + (−0.00118 + 0.000347i)19-s + (0.523 + 3.64i)21-s + (4.78 + 0.383i)23-s + (−1.75 − 12.2i)25-s + (−4.69 + 1.37i)27-s + (1.13 + 0.333i)29-s + (−1.36 − 2.99i)31-s + ⋯ |
| L(s) = 1 | + (−0.436 + 0.955i)3-s + (−1.21 + 1.40i)5-s + (0.643 − 0.413i)7-s + (−0.0681 − 0.0786i)9-s + (0.148 − 1.03i)11-s + (1.37 + 0.883i)13-s + (−0.813 − 1.78i)15-s + (−0.596 − 0.175i)17-s + (−0.000271 + 7.97e−5i)19-s + (0.114 + 0.795i)21-s + (0.996 + 0.0800i)23-s + (−0.351 − 2.44i)25-s + (−0.903 + 0.265i)27-s + (0.211 + 0.0619i)29-s + (−0.245 − 0.538i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0259 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0259 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.553131 + 0.567653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.553131 + 0.567653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (-4.78 - 0.383i)T \) |
| good | 3 | \( 1 + (0.756 - 1.65i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (2.72 - 3.14i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.70 + 1.09i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.491 + 3.41i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-4.95 - 3.18i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (2.45 + 0.722i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.00118 - 0.000347i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.13 - 0.333i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.36 + 2.99i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.23 - 1.42i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (2.51 - 2.89i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.545 + 1.19i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 + (-4.08 + 2.62i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (6.64 + 4.27i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (2.19 + 4.81i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.339 + 2.36i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.200 - 1.39i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.30 + 1.85i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (14.8 + 9.51i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (6.67 + 7.70i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.33 + 11.6i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.50 + 7.50i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−14.50184716868446316506959559139, −13.57100503617519522116669135192, −11.45461799360801505792744865165, −11.20180248532526545240415933845, −10.50456968231625062915940927194, −8.807131794250213851278696230647, −7.52792168377472146486627488867, −6.30848016570242652977631119384, −4.43571660280964110668204731197, −3.48230188305601206118291930612,
1.22216756229914468491495340579, 4.17651643449771260480765615458, 5.46632332746696099202642608604, 7.11061138617096054259436989633, 8.165776052961200877920014682319, 8.984709046993786981274371946281, 10.98771588069493302936339336254, 11.97055914210059986833355706427, 12.59822057817779784556887462885, 13.31542056368835456852153940743
