| L(s) = 1 | + (−26.6 − 17.1i)2-s + (−29.0 − 202. i)3-s + (204. + 447. i)4-s + (210. + 61.8i)5-s + (−2.69e3 + 5.89e3i)6-s + (−3.08e3 + 3.56e3i)7-s + (−90.4 + 629. i)8-s + (−2.11e4 + 6.22e3i)9-s + (−4.55e3 − 5.25e3i)10-s + (−1.82e4 + 1.17e4i)11-s + (8.45e4 − 5.43e4i)12-s + (3.58e4 + 4.13e4i)13-s + (1.43e5 − 4.20e4i)14-s + (6.38e3 − 4.44e4i)15-s + (1.78e5 − 2.05e5i)16-s + (−1.58e4 + 3.46e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.17 − 0.757i)2-s + (−0.207 − 1.44i)3-s + (0.399 + 0.873i)4-s + (0.150 + 0.0442i)5-s + (−0.847 + 1.85i)6-s + (−0.485 + 0.560i)7-s + (−0.00780 + 0.0543i)8-s + (−1.07 + 0.316i)9-s + (−0.144 − 0.166i)10-s + (−0.375 + 0.241i)11-s + (1.17 − 0.756i)12-s + (0.348 + 0.401i)13-s + (0.997 − 0.292i)14-s + (0.0325 − 0.226i)15-s + (0.679 − 0.784i)16-s + (−0.0458 + 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0545i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.416379 + 0.0113737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.416379 + 0.0113737i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (2.09e5 + 1.32e6i)T \) |
| good | 2 | \( 1 + (26.6 + 17.1i)T + (212. + 465. i)T^{2} \) |
| 3 | \( 1 + (29.0 + 202. i)T + (-1.88e4 + 5.54e3i)T^{2} \) |
| 5 | \( 1 + (-210. - 61.8i)T + (1.64e6 + 1.05e6i)T^{2} \) |
| 7 | \( 1 + (3.08e3 - 3.56e3i)T + (-5.74e6 - 3.99e7i)T^{2} \) |
| 11 | \( 1 + (1.82e4 - 1.17e4i)T + (9.79e8 - 2.14e9i)T^{2} \) |
| 13 | \( 1 + (-3.58e4 - 4.13e4i)T + (-1.50e9 + 1.04e10i)T^{2} \) |
| 17 | \( 1 + (1.58e4 - 3.46e4i)T + (-7.76e10 - 8.96e10i)T^{2} \) |
| 19 | \( 1 + (-2.17e5 - 4.77e5i)T + (-2.11e11 + 2.43e11i)T^{2} \) |
| 29 | \( 1 + (4.91e5 - 1.07e6i)T + (-9.50e12 - 1.09e13i)T^{2} \) |
| 31 | \( 1 + (1.12e6 - 7.81e6i)T + (-2.53e13 - 7.44e12i)T^{2} \) |
| 37 | \( 1 + (2.30e6 - 6.75e5i)T + (1.09e14 - 7.02e13i)T^{2} \) |
| 41 | \( 1 + (-2.19e7 - 6.43e6i)T + (2.75e14 + 1.76e14i)T^{2} \) |
| 43 | \( 1 + (-2.13e6 - 1.48e7i)T + (-4.82e14 + 1.41e14i)T^{2} \) |
| 47 | \( 1 - 3.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (1.10e7 - 1.27e7i)T + (-4.69e14 - 3.26e15i)T^{2} \) |
| 59 | \( 1 + (-3.38e7 - 3.90e7i)T + (-1.23e15 + 8.57e15i)T^{2} \) |
| 61 | \( 1 + (2.09e7 - 1.45e8i)T + (-1.12e16 - 3.29e15i)T^{2} \) |
| 67 | \( 1 + (2.30e8 + 1.48e8i)T + (1.13e16 + 2.47e16i)T^{2} \) |
| 71 | \( 1 + (1.02e8 + 6.57e7i)T + (1.90e16 + 4.17e16i)T^{2} \) |
| 73 | \( 1 + (-1.98e8 - 4.34e8i)T + (-3.85e16 + 4.44e16i)T^{2} \) |
| 79 | \( 1 + (3.12e8 + 3.60e8i)T + (-1.70e16 + 1.18e17i)T^{2} \) |
| 83 | \( 1 + (-2.76e8 + 8.10e7i)T + (1.57e17 - 1.01e17i)T^{2} \) |
| 89 | \( 1 + (2.58e7 + 1.80e8i)T + (-3.36e17 + 9.87e16i)T^{2} \) |
| 97 | \( 1 + (-1.34e9 - 3.94e8i)T + (6.39e17 + 4.11e17i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.24791219916262060910060898953, −14.16341582573497374305174132974, −12.66398410885157575742983558468, −11.87405592412072645035979098701, −10.38217592374025075173010223385, −8.900249255743260817367341981613, −7.62988974038677597427560309097, −6.06016792414564449701170235744, −2.48764780071086893445792690153, −1.25458753903683088702951031376,
0.30794099299010809377490446931, 3.76614987647627621018791608161, 5.71650787270839431669452505713, 7.53311443971251261465790613015, 9.172823942016566623470336285641, 9.942803289437206398656640705934, 11.04911037193166443034735089285, 13.39390311976453562920976509808, 15.30436079935184876688623403817, 15.88712721642794157768764846398
