| L(s) = 1 | + (0.306 − 2.12i)2-s + (3.96 + 8.68i)3-s + (3.23 + 0.951i)4-s + (−11.4 − 13.2i)5-s + (19.7 − 5.78i)6-s + (−2.65 − 1.70i)7-s + (10.1 − 22.2i)8-s + (−42.0 + 48.5i)9-s + (−31.6 + 20.3i)10-s + (−1.83 − 12.7i)11-s + (4.59 + 31.9i)12-s + (2.06 − 1.32i)13-s + (−4.43 + 5.12i)14-s + (69.4 − 152. i)15-s + (−21.5 − 13.8i)16-s + (−60.1 + 17.6i)17-s + ⋯ |
| L(s) = 1 | + (0.108 − 0.752i)2-s + (0.763 + 1.67i)3-s + (0.404 + 0.118i)4-s + (−1.02 − 1.18i)5-s + (1.34 − 0.393i)6-s + (−0.143 − 0.0920i)7-s + (0.449 − 0.983i)8-s + (−1.55 + 1.79i)9-s + (−1.00 + 0.643i)10-s + (−0.0502 − 0.349i)11-s + (0.110 + 0.767i)12-s + (0.0440 − 0.0283i)13-s + (−0.0847 + 0.0977i)14-s + (1.19 − 2.61i)15-s + (−0.336 − 0.216i)16-s + (−0.857 + 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00415i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.37946 - 0.00286560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37946 - 0.00286560i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-41.6 - 102. i)T \) |
| good | 2 | \( 1 + (-0.306 + 2.12i)T + (-7.67 - 2.25i)T^{2} \) |
| 3 | \( 1 + (-3.96 - 8.68i)T + (-17.6 + 20.4i)T^{2} \) |
| 5 | \( 1 + (11.4 + 13.2i)T + (-17.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (2.65 + 1.70i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (1.83 + 12.7i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (-2.06 + 1.32i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (60.1 - 17.6i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (-38.0 - 11.1i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (-179. + 52.8i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-1.30 + 2.86i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (86.6 - 99.9i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (35.4 + 40.9i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-97.9 - 214. i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 + 128.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-159. - 102. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (423. - 272. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (19.9 - 43.6i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-69.8 + 485. i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-141. + 984. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-225. - 66.2i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-1.12e3 + 723. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (699. - 807. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (551. + 1.20e3i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-737. - 851. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−16.76156521119633278397387452715, −15.92739753051420473130622766259, −15.33930903585945150393712263749, −13.45643183292775635342885842782, −11.87369488916140146112660225746, −10.75453785281900964416001377440, −9.370598055330833839357917879709, −8.128999346886437796702792044000, −4.62066261734217924091889554817, −3.41677382049646695485498215623,
2.67523203914192062256226371653, 6.61429591643192235510662376392, 7.19909015449605403304713103403, 8.331568635558544433472278347649, 11.11955549198195854701768227613, 12.30368009519053455228150243769, 13.92033165388910203826743435669, 14.76187486394602118153972694737, 15.74644505712104588801920061918, 17.66150156063888885519861289794
