| L(s) = 1 | + (0.0282 − 0.196i)2-s + (−3.66 − 8.01i)3-s + (7.63 + 2.24i)4-s + (−6.59 − 7.60i)5-s + (−1.67 + 0.493i)6-s + (21.3 + 13.7i)7-s + (1.31 − 2.88i)8-s + (−33.1 + 38.2i)9-s + (−1.68 + 1.08i)10-s + (2.01 + 14.0i)11-s + (−9.98 − 69.4i)12-s + (−3.85 + 2.47i)13-s + (3.29 − 3.80i)14-s + (−36.8 + 80.6i)15-s + (53.0 + 34.0i)16-s + (98.0 − 28.7i)17-s + ⋯ |
| L(s) = 1 | + (0.00999 − 0.0695i)2-s + (−0.704 − 1.54i)3-s + (0.954 + 0.280i)4-s + (−0.589 − 0.680i)5-s + (−0.114 + 0.0335i)6-s + (1.15 + 0.739i)7-s + (0.0582 − 0.127i)8-s + (−1.22 + 1.41i)9-s + (−0.0532 + 0.0341i)10-s + (0.0552 + 0.384i)11-s + (−0.240 − 1.67i)12-s + (−0.0822 + 0.0528i)13-s + (0.0629 − 0.0726i)14-s + (−0.634 + 1.38i)15-s + (0.828 + 0.532i)16-s + (1.39 − 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.876567 - 0.637009i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.876567 - 0.637009i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (83.9 - 71.5i)T \) |
| good | 2 | \( 1 + (-0.0282 + 0.196i)T + (-7.67 - 2.25i)T^{2} \) |
| 3 | \( 1 + (3.66 + 8.01i)T + (-17.6 + 20.4i)T^{2} \) |
| 5 | \( 1 + (6.59 + 7.60i)T + (-17.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-21.3 - 13.7i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (-2.01 - 14.0i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (3.85 - 2.47i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-98.0 + 28.7i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (66.4 + 19.5i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (202. - 59.4i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (117. - 256. i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-114. + 131. i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (66.9 + 77.2i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (-53.4 - 116. i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 - 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (81.5 + 52.3i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-232. + 149. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (-190. + 416. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-49.8 + 346. i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (109. - 758. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (869. + 255. i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-377. + 242. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (281. - 325. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (138. + 304. i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (557. + 643. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−17.26056155001359908029069293332, −16.15237705714742371341247459257, −14.57264011556464627456723039742, −12.62114664500854587978851585014, −12.06898481777303903534359631489, −11.20004020622204992879477729570, −8.209788716578022339569693349357, −7.28361987943604452994669976214, −5.55177791440571751632964091513, −1.73584609366888119606850895528,
3.93850361285048538275506321435, 5.76861665654945848722486832347, 7.73291008711341703384675052604, 10.16281833463862138104411933134, 10.96485744848147385369296164470, 11.67141840969087897045557356030, 14.67039533803447125085465009733, 15.01477431204622249348586095311, 16.45100993370595708075756729466, 17.04254317661647607923942580411
