| L(s) = 1 | + (−0.348 − 0.762i)3-s + (2.89 − 2.51i)5-s + (1.34 − 2.09i)7-s + (5.43 − 6.27i)9-s + (1.56 − 0.225i)11-s + (6.41 − 4.12i)13-s + (−2.92 − 1.33i)15-s + (0.0676 + 0.230i)17-s + (−4.01 + 13.6i)19-s + (−2.06 − 0.297i)21-s + (−19.6 + 11.9i)23-s + (−1.46 + 10.1i)25-s + (−13.9 − 4.08i)27-s + (1.28 − 0.377i)29-s + (−12.5 + 27.4i)31-s + ⋯ |
| L(s) = 1 | + (−0.116 − 0.254i)3-s + (0.579 − 0.502i)5-s + (0.192 − 0.299i)7-s + (0.603 − 0.696i)9-s + (0.142 − 0.0205i)11-s + (0.493 − 0.317i)13-s + (−0.194 − 0.0890i)15-s + (0.00397 + 0.0135i)17-s + (−0.211 + 0.719i)19-s + (−0.0984 − 0.0141i)21-s + (−0.854 + 0.519i)23-s + (−0.0586 + 0.407i)25-s + (−0.515 − 0.151i)27-s + (0.0443 − 0.0130i)29-s + (−0.404 + 0.886i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31580 - 0.519829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31580 - 0.519829i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (19.6 - 11.9i)T \) |
| good | 3 | \( 1 + (0.348 + 0.762i)T + (-5.89 + 6.80i)T^{2} \) |
| 5 | \( 1 + (-2.89 + 2.51i)T + (3.55 - 24.7i)T^{2} \) |
| 7 | \( 1 + (-1.34 + 2.09i)T + (-20.3 - 44.5i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 0.225i)T + (116. - 34.0i)T^{2} \) |
| 13 | \( 1 + (-6.41 + 4.12i)T + (70.2 - 153. i)T^{2} \) |
| 17 | \( 1 + (-0.0676 - 0.230i)T + (-243. + 156. i)T^{2} \) |
| 19 | \( 1 + (4.01 - 13.6i)T + (-303. - 195. i)T^{2} \) |
| 29 | \( 1 + (-1.28 + 0.377i)T + (707. - 454. i)T^{2} \) |
| 31 | \( 1 + (12.5 - 27.4i)T + (-629. - 726. i)T^{2} \) |
| 37 | \( 1 + (-17.4 - 15.0i)T + (194. + 1.35e3i)T^{2} \) |
| 41 | \( 1 + (7.89 + 9.11i)T + (-239. + 1.66e3i)T^{2} \) |
| 43 | \( 1 + (33.1 - 15.1i)T + (1.21e3 - 1.39e3i)T^{2} \) |
| 47 | \( 1 + 1.41T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-36.5 + 56.8i)T + (-1.16e3 - 2.55e3i)T^{2} \) |
| 59 | \( 1 + (-48.6 + 31.2i)T + (1.44e3 - 3.16e3i)T^{2} \) |
| 61 | \( 1 + (-53.6 - 24.4i)T + (2.43e3 + 2.81e3i)T^{2} \) |
| 67 | \( 1 + (2.86 + 0.411i)T + (4.30e3 + 1.26e3i)T^{2} \) |
| 71 | \( 1 + (-9.59 + 66.7i)T + (-4.83e3 - 1.42e3i)T^{2} \) |
| 73 | \( 1 + (57.8 + 16.9i)T + (4.48e3 + 2.88e3i)T^{2} \) |
| 79 | \( 1 + (-60.6 - 94.2i)T + (-2.59e3 + 5.67e3i)T^{2} \) |
| 83 | \( 1 + (19.9 + 17.2i)T + (980. + 6.81e3i)T^{2} \) |
| 89 | \( 1 + (126. - 57.9i)T + (5.18e3 - 5.98e3i)T^{2} \) |
| 97 | \( 1 + (101. - 87.8i)T + (1.33e3 - 9.31e3i)T^{2} \) |
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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
−13.53257865639038947470937898599, −12.73333827157925217951070002303, −11.70297138110397600285443213906, −10.32705969015219007959303829893, −9.364669962966576869840584977232, −8.090466113649587165752517766322, −6.70657635105983216227126129234, −5.49652205070736990569436475676, −3.84341333918620177385650767275, −1.43463271984225639172109511532,
2.21948959841827735135531449953, 4.27073659747190790981262852546, 5.75280362091500072104844770445, 7.00147359014371180148426596643, 8.432546240637303076772842828087, 9.751657692872612164329215674230, 10.63652366611072181104519984406, 11.67122796328635013517274929009, 13.05515471694657185956622344359, 13.91247501499833124796540259582
