| L(s) = 1 | + (0.496 − 3.45i)2-s + (1.53 + 3.36i)3-s + (−7.83 − 2.30i)4-s + (−4.84 + 4.19i)5-s + (12.3 − 3.63i)6-s + (3.55 − 5.52i)7-s + (−6.04 + 13.2i)8-s + (−3.07 + 3.55i)9-s + (12.0 + 18.8i)10-s + (−7.17 + 1.03i)11-s + (−4.30 − 29.9i)12-s + (9.96 − 6.40i)13-s + (−17.3 − 15.0i)14-s + (−21.5 − 9.85i)15-s + (15.1 + 9.76i)16-s + (1.04 + 3.55i)17-s + ⋯ |
| L(s) = 1 | + (0.248 − 1.72i)2-s + (0.512 + 1.12i)3-s + (−1.95 − 0.575i)4-s + (−0.968 + 0.839i)5-s + (2.06 − 0.606i)6-s + (0.507 − 0.789i)7-s + (−0.755 + 1.65i)8-s + (−0.342 + 0.394i)9-s + (1.20 + 1.88i)10-s + (−0.652 + 0.0937i)11-s + (−0.358 − 2.49i)12-s + (0.766 − 0.492i)13-s + (−1.23 − 1.07i)14-s + (−1.43 − 0.656i)15-s + (0.949 + 0.610i)16-s + (0.0614 + 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.394 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.817012 - 0.538671i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.817012 - 0.538671i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (21.0 - 9.35i)T \) |
| good | 2 | \( 1 + (-0.496 + 3.45i)T + (-3.83 - 1.12i)T^{2} \) |
| 3 | \( 1 + (-1.53 - 3.36i)T + (-5.89 + 6.80i)T^{2} \) |
| 5 | \( 1 + (4.84 - 4.19i)T + (3.55 - 24.7i)T^{2} \) |
| 7 | \( 1 + (-3.55 + 5.52i)T + (-20.3 - 44.5i)T^{2} \) |
| 11 | \( 1 + (7.17 - 1.03i)T + (116. - 34.0i)T^{2} \) |
| 13 | \( 1 + (-9.96 + 6.40i)T + (70.2 - 153. i)T^{2} \) |
| 17 | \( 1 + (-1.04 - 3.55i)T + (-243. + 156. i)T^{2} \) |
| 19 | \( 1 + (3.34 - 11.4i)T + (-303. - 195. i)T^{2} \) |
| 29 | \( 1 + (-44.1 + 12.9i)T + (707. - 454. i)T^{2} \) |
| 31 | \( 1 + (-11.7 + 25.7i)T + (-629. - 726. i)T^{2} \) |
| 37 | \( 1 + (-1.87 - 1.62i)T + (194. + 1.35e3i)T^{2} \) |
| 41 | \( 1 + (26.9 + 31.1i)T + (-239. + 1.66e3i)T^{2} \) |
| 43 | \( 1 + (44.3 - 20.2i)T + (1.21e3 - 1.39e3i)T^{2} \) |
| 47 | \( 1 + 36.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-3.67 + 5.71i)T + (-1.16e3 - 2.55e3i)T^{2} \) |
| 59 | \( 1 + (42.6 - 27.3i)T + (1.44e3 - 3.16e3i)T^{2} \) |
| 61 | \( 1 + (-88.9 - 40.6i)T + (2.43e3 + 2.81e3i)T^{2} \) |
| 67 | \( 1 + (28.0 + 4.03i)T + (4.30e3 + 1.26e3i)T^{2} \) |
| 71 | \( 1 + (-4.93 + 34.3i)T + (-4.83e3 - 1.42e3i)T^{2} \) |
| 73 | \( 1 + (7.15 + 2.09i)T + (4.48e3 + 2.88e3i)T^{2} \) |
| 79 | \( 1 + (71.3 + 110. i)T + (-2.59e3 + 5.67e3i)T^{2} \) |
| 83 | \( 1 + (-16.5 - 14.3i)T + (980. + 6.81e3i)T^{2} \) |
| 89 | \( 1 + (-145. + 66.3i)T + (5.18e3 - 5.98e3i)T^{2} \) |
| 97 | \( 1 + (81.7 - 70.8i)T + (1.33e3 - 9.31e3i)T^{2} \) |
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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
−17.97497636505737185115338702079, −15.83931649061036516814283224474, −14.71522913078469316664587527796, −13.53546705425950027742557181206, −11.81040913822049062189249032247, −10.66134753858708199034353863336, −10.10381796001147887348709899479, −8.170143729664327521446751818493, −4.32560642839855248715965469750, −3.32756736453557561336141407821,
4.86569431530740825682986597350, 6.71323233649020210521627211278, 8.251718659698555867406770865099, 8.450120030841440590909398683258, 12.02649871119482116597693614097, 13.17527900532554603443111635920, 14.22582694153667519409292668647, 15.59283660830610838795184461520, 16.24885623036844045924752718303, 17.87906934237354596933665154997
