| L(s) = 1 | + (−1.35 − 0.406i)2-s + (−0.128 − 0.437i)3-s + (1.66 + 1.10i)4-s + (0.782 − 1.21i)5-s + (−0.00376 + 0.645i)6-s + (−0.0556 − 0.387i)7-s + (−1.81 − 2.16i)8-s + (2.34 − 1.50i)9-s + (−1.55 + 1.33i)10-s + (1.91 − 4.18i)11-s + (0.267 − 0.872i)12-s + (−0.355 + 2.47i)13-s + (−0.0819 + 0.547i)14-s + (−0.633 − 0.185i)15-s + (1.57 + 3.67i)16-s + (−3.28 + 2.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.957 − 0.287i)2-s + (−0.0741 − 0.252i)3-s + (0.834 + 0.550i)4-s + (0.349 − 0.544i)5-s + (−0.00153 + 0.263i)6-s + (−0.0210 − 0.146i)7-s + (−0.641 − 0.767i)8-s + (0.782 − 0.503i)9-s + (−0.491 + 0.420i)10-s + (0.576 − 1.26i)11-s + (0.0771 − 0.251i)12-s + (−0.0987 + 0.686i)13-s + (−0.0218 + 0.146i)14-s + (−0.163 − 0.0480i)15-s + (0.394 + 0.919i)16-s + (−0.797 + 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.633042 - 0.324155i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.633042 - 0.324155i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 + 0.406i)T \) |
| 23 | \( 1 + (-0.925 - 4.70i)T \) |
| good | 3 | \( 1 + (0.128 + 0.437i)T + (-2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (-0.782 + 1.21i)T + (-2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.0556 + 0.387i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.91 + 4.18i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.355 - 2.47i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (3.28 - 2.84i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.65 - 1.90i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (0.799 + 0.922i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.797 - 2.71i)T + (-26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (1.21 + 1.89i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (8.05 + 5.17i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.19 + 0.938i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + (1.64 - 0.236i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-9.43 - 1.35i)T + (56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (1.72 - 5.86i)T + (-51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (6.20 + 13.5i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (12.4 - 5.70i)T + (46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (1.04 - 1.20i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.43 - 9.99i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-10.8 + 6.95i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.00 - 13.6i)T + (-74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-2.26 + 3.52i)T + (-40.2 - 88.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.66372643324960362939233532318, −12.68198262028596373420360440111, −11.66753805928097876622536729502, −10.61884233565934704179590513221, −9.337072322384986892529878559458, −8.657675464280332839208589169847, −7.17494644819501720564477857589, −6.07845306335042419689616617194, −3.80628651300063517832925021887, −1.49035388028059484236987394892,
2.28143666750605414929842905188, 4.81052853317218680446852327276, 6.54832077726884301292321495632, 7.35544705304208457031938982618, 8.836310719742867495677759705554, 9.983867429427435637033610297751, 10.56134431036526378325179020968, 11.85124967987055203821535061658, 13.17338712805113195011956568217, 14.67618524210137956684852272507
