| L(s) = 1 | + (1.44 − 0.424i)2-s + (−0.590 − 0.680i)3-s + (−1.45 + 0.934i)4-s + (−1.77 + 0.255i)5-s + (−1.14 − 0.734i)6-s + (2.68 + 1.22i)7-s + (−5.65 + 6.52i)8-s + (1.16 − 8.10i)9-s + (−2.46 + 1.12i)10-s + (0.324 − 1.10i)11-s + (1.49 + 0.438i)12-s + (0.859 + 1.88i)13-s + (4.40 + 0.633i)14-s + (1.22 + 1.06i)15-s + (−2.53 + 5.55i)16-s + (12.8 − 19.9i)17-s + ⋯ |
| L(s) = 1 | + (0.723 − 0.212i)2-s + (−0.196 − 0.226i)3-s + (−0.363 + 0.233i)4-s + (−0.355 + 0.0511i)5-s + (−0.190 − 0.122i)6-s + (0.383 + 0.175i)7-s + (−0.706 + 0.815i)8-s + (0.129 − 0.900i)9-s + (−0.246 + 0.112i)10-s + (0.0295 − 0.100i)11-s + (0.124 + 0.0365i)12-s + (0.0661 + 0.144i)13-s + (0.314 + 0.0452i)14-s + (0.0815 + 0.0706i)15-s + (−0.158 + 0.346i)16-s + (0.754 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.04176 - 0.115117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04176 - 0.115117i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (22.9 + 1.37i)T \) |
| good | 2 | \( 1 + (-1.44 + 0.424i)T + (3.36 - 2.16i)T^{2} \) |
| 3 | \( 1 + (0.590 + 0.680i)T + (-1.28 + 8.90i)T^{2} \) |
| 5 | \( 1 + (1.77 - 0.255i)T + (23.9 - 7.04i)T^{2} \) |
| 7 | \( 1 + (-2.68 - 1.22i)T + (32.0 + 37.0i)T^{2} \) |
| 11 | \( 1 + (-0.324 + 1.10i)T + (-101. - 65.4i)T^{2} \) |
| 13 | \( 1 + (-0.859 - 1.88i)T + (-110. + 127. i)T^{2} \) |
| 17 | \( 1 + (-12.8 + 19.9i)T + (-120. - 262. i)T^{2} \) |
| 19 | \( 1 + (-16.8 - 26.2i)T + (-149. + 328. i)T^{2} \) |
| 29 | \( 1 + (13.4 + 8.65i)T + (349. + 765. i)T^{2} \) |
| 31 | \( 1 + (25.8 - 29.7i)T + (-136. - 951. i)T^{2} \) |
| 37 | \( 1 + (-59.3 - 8.53i)T + (1.31e3 + 385. i)T^{2} \) |
| 41 | \( 1 + (3.46 + 24.0i)T + (-1.61e3 + 473. i)T^{2} \) |
| 43 | \( 1 + (-6.13 + 5.31i)T + (263. - 1.83e3i)T^{2} \) |
| 47 | \( 1 - 58.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + (36.3 + 16.6i)T + (1.83e3 + 2.12e3i)T^{2} \) |
| 59 | \( 1 + (-19.0 - 41.7i)T + (-2.27e3 + 2.63e3i)T^{2} \) |
| 61 | \( 1 + (-42.6 - 36.9i)T + (529. + 3.68e3i)T^{2} \) |
| 67 | \( 1 + (22.4 + 76.5i)T + (-3.77e3 + 2.42e3i)T^{2} \) |
| 71 | \( 1 + (58.8 - 17.2i)T + (4.24e3 - 2.72e3i)T^{2} \) |
| 73 | \( 1 + (-80.6 + 51.8i)T + (2.21e3 - 4.84e3i)T^{2} \) |
| 79 | \( 1 + (-19.8 + 9.05i)T + (4.08e3 - 4.71e3i)T^{2} \) |
| 83 | \( 1 + (-59.7 - 8.59i)T + (6.60e3 + 1.94e3i)T^{2} \) |
| 89 | \( 1 + (52.9 - 45.8i)T + (1.12e3 - 7.84e3i)T^{2} \) |
| 97 | \( 1 + (83.0 - 11.9i)T + (9.02e3 - 2.65e3i)T^{2} \) |
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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
−17.87133957421723456568824758289, −16.33260205434929697196775701224, −14.78680582450645105578039902774, −13.82080361468241694309877067007, −12.27959728766015304732282872436, −11.68505638783037680409911153035, −9.477618758638267987107619138920, −7.78289322023017320713655507147, −5.62983546690853782426384080275, −3.71539445484542069488124611256,
4.22323033725125680272881102627, 5.66799309611214080007374264712, 7.81773480421544286083125359818, 9.729991637754150206926398507721, 11.23630765385808139432898026050, 12.83194099127088298920040808342, 13.93883774949076036462839748538, 15.10677953159637770723398832683, 16.21272647089582150461702723818, 17.66054105977335365903287743954
