| 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
−17.66150156063888885519861289794, −15.74644505712104588801920061918, −14.76187486394602118153972694737, −13.92033165388910203826743435669, −12.30368009519053455228150243769, −11.11955549198195854701768227613, −8.331568635558544433472278347649, −7.19909015449605403304713103403, −6.61429591643192235510662376392, −2.67523203914192062256226371653,
3.41677382049646695485498215623, 4.62066261734217924091889554817, 8.128999346886437796702792044000, 9.370598055330833839357917879709, 10.75453785281900964416001377440, 11.87369488916140146112660225746, 13.45643183292775635342885842782, 15.33930903585945150393712263749, 15.92739753051420473130622766259, 16.76156521119633278397387452715
