| 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
−17.70203425978957641768717895494, −17.04723044924743086585196461046, −15.57526305208137422104624284769, −15.00562091769965968947725002073, −12.52410114655400166693330577158, −11.70197430633765723113015228369, −9.811646528512427225180686061796, −7.87914287964310171311402966194, −6.64194069350663578132544817298, −5.45606808982273237483720539487,
0.53231084436137893030156383113, 4.12288368997831282559911382376, 6.39159201211161684177390000845, 8.972375914314242561383084718163, 10.44279477694412094117189764656, 11.42582646159261030681269081679, 11.97045960279196273071879694915, 13.88195261514692020456066224824, 16.21254614123755218935202707659, 16.73849276215109129054427254769
