| L(s) = 1 | + (−1.58 − 3.46i)2-s + (−7.04 + 2.06i)3-s + (−4.26 + 4.92i)4-s + (−4.88 − 3.14i)5-s + (18.3 + 21.1i)6-s + (−2.25 − 15.6i)7-s + (−5.44 − 1.59i)8-s + (22.6 − 14.5i)9-s + (−3.14 + 21.9i)10-s + (27.3 − 59.8i)11-s + (19.8 − 43.4i)12-s + (−10.3 + 72.0i)13-s + (−50.7 + 32.5i)14-s + (40.9 + 12.0i)15-s + (10.4 + 72.9i)16-s + (−25.6 − 29.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.559 − 1.22i)2-s + (−1.35 + 0.398i)3-s + (−0.532 + 0.615i)4-s + (−0.437 − 0.280i)5-s + (1.24 + 1.43i)6-s + (−0.121 − 0.845i)7-s + (−0.240 − 0.0706i)8-s + (0.838 − 0.538i)9-s + (−0.0996 + 0.692i)10-s + (0.749 − 1.64i)11-s + (0.477 − 1.04i)12-s + (−0.221 + 1.53i)13-s + (−0.968 + 0.622i)14-s + (0.704 + 0.206i)15-s + (0.163 + 1.13i)16-s + (−0.366 − 0.422i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0701i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0126503 + 0.360170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0126503 + 0.360170i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (90.0 + 63.7i)T \) |
| good | 2 | \( 1 + (1.58 + 3.46i)T + (-5.23 + 6.04i)T^{2} \) |
| 3 | \( 1 + (7.04 - 2.06i)T + (22.7 - 14.5i)T^{2} \) |
| 5 | \( 1 + (4.88 + 3.14i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (2.25 + 15.6i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (-27.3 + 59.8i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (10.3 - 72.0i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (25.6 + 29.6i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (-25.7 + 29.7i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (-76.4 - 88.2i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (44.2 + 12.9i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-232. + 149. i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (221. + 142. i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (158. - 46.5i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 - 168.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-5.17 - 35.9i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (74.9 - 521. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-177. - 52.2i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (5.44 + 11.9i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (40.5 + 88.7i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (-283. + 326. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-148. + 1.03e3i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-865. + 556. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (896. - 263. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-438. - 281. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−16.73849276215109129054427254769, −16.21254614123755218935202707659, −13.88195261514692020456066224824, −11.97045960279196273071879694915, −11.42582646159261030681269081679, −10.44279477694412094117189764656, −8.972375914314242561383084718163, −6.39159201211161684177390000845, −4.12288368997831282559911382376, −0.53231084436137893030156383113,
5.45606808982273237483720539487, 6.64194069350663578132544817298, 7.87914287964310171311402966194, 9.811646528512427225180686061796, 11.70197430633765723113015228369, 12.52410114655400166693330577158, 15.00562091769965968947725002073, 15.57526305208137422104624284769, 17.04723044924743086585196461046, 17.70203425978957641768717895494
