| L(s) = 1 | + (−1.38 − 1.59i)2-s + (−3.80 − 2.44i)3-s + (−0.0656 + 0.456i)4-s + (6.26 + 2.85i)5-s + (1.35 + 9.45i)6-s + (2.54 − 8.65i)7-s + (−6.28 + 4.04i)8-s + (4.75 + 10.4i)9-s + (−4.09 − 13.9i)10-s + (5.67 + 4.91i)11-s + (1.36 − 1.57i)12-s + (1.51 − 0.444i)13-s + (−17.3 + 7.91i)14-s + (−16.8 − 26.1i)15-s + (16.9 + 4.96i)16-s + (−2.21 + 0.318i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.798i)2-s + (−1.26 − 0.814i)3-s + (−0.0164 + 0.114i)4-s + (1.25 + 0.571i)5-s + (0.226 + 1.57i)6-s + (0.363 − 1.23i)7-s + (−0.785 + 0.505i)8-s + (0.527 + 1.15i)9-s + (−0.409 − 1.39i)10-s + (0.515 + 0.447i)11-s + (0.113 − 0.131i)12-s + (0.116 − 0.0342i)13-s + (−1.23 + 0.565i)14-s + (−1.12 − 1.74i)15-s + (1.05 + 0.310i)16-s + (−0.130 + 0.0187i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.313339 - 0.467754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.313339 - 0.467754i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-6.76 - 21.9i)T \) |
| good | 2 | \( 1 + (1.38 + 1.59i)T + (-0.569 + 3.95i)T^{2} \) |
| 3 | \( 1 + (3.80 + 2.44i)T + (3.73 + 8.18i)T^{2} \) |
| 5 | \( 1 + (-6.26 - 2.85i)T + (16.3 + 18.8i)T^{2} \) |
| 7 | \( 1 + (-2.54 + 8.65i)T + (-41.2 - 26.4i)T^{2} \) |
| 11 | \( 1 + (-5.67 - 4.91i)T + (17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-1.51 + 0.444i)T + (142. - 91.3i)T^{2} \) |
| 17 | \( 1 + (2.21 - 0.318i)T + (277. - 81.4i)T^{2} \) |
| 19 | \( 1 + (-8.82 - 1.26i)T + (346. + 101. i)T^{2} \) |
| 29 | \( 1 + (4.03 + 28.0i)T + (-806. + 236. i)T^{2} \) |
| 31 | \( 1 + (40.2 - 25.8i)T + (399. - 874. i)T^{2} \) |
| 37 | \( 1 + (-31.8 + 14.5i)T + (896. - 1.03e3i)T^{2} \) |
| 41 | \( 1 + (9.80 - 21.4i)T + (-1.10e3 - 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-16.4 + 25.5i)T + (-768. - 1.68e3i)T^{2} \) |
| 47 | \( 1 + 62.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + (7.57 - 25.8i)T + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (-44.5 + 13.0i)T + (2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-41.2 - 64.1i)T + (-1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (42.4 - 36.7i)T + (638. - 4.44e3i)T^{2} \) |
| 71 | \( 1 + (41.1 + 47.4i)T + (-717. + 4.98e3i)T^{2} \) |
| 73 | \( 1 + (9.65 - 67.1i)T + (-5.11e3 - 1.50e3i)T^{2} \) |
| 79 | \( 1 + (18.1 + 61.7i)T + (-5.25e3 + 3.37e3i)T^{2} \) |
| 83 | \( 1 + (51.8 - 23.6i)T + (4.51e3 - 5.20e3i)T^{2} \) |
| 89 | \( 1 + (25.6 - 39.8i)T + (-3.29e3 - 7.20e3i)T^{2} \) |
| 97 | \( 1 + (112. + 51.3i)T + (6.16e3 + 7.11e3i)T^{2} \) |
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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
−17.72582996052307561432126031680, −16.88502796917604461632357942826, −14.44258418726199525371280719992, −13.19775952555330102801980688754, −11.60703771056307806026666795662, −10.72381577479829092112324120196, −9.696090886471768696510023291548, −7.09804639594227822977154384392, −5.75572916446322716921709034816, −1.46813513815667992899871326013,
5.32971370510433500814029524432, 6.25046989440639473602576980442, 8.773066199674099715397114750120, 9.673178327808036226137985514018, 11.40060053173278964420910360836, 12.69949433113958177079206539846, 14.80748018756301867845488854947, 16.16608508643368501278461297387, 16.75955248543794914407373064218, 17.72181540099004643191007721788
