| L(s) = 1 | + (−0.881 − 1.01i)2-s + (2.64 + 1.70i)3-s + (0.311 − 2.16i)4-s + (−2.08 − 0.951i)5-s + (−0.602 − 4.19i)6-s + (−2.90 + 9.89i)7-s + (−7.00 + 4.50i)8-s + (0.375 + 0.822i)9-s + (0.867 + 2.95i)10-s + (−6.07 − 5.26i)11-s + (4.51 − 5.20i)12-s + (14.4 − 4.25i)13-s + (12.6 − 5.76i)14-s + (−3.89 − 6.06i)15-s + (2.34 + 0.689i)16-s + (−13.1 + 1.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.440 − 0.508i)2-s + (0.882 + 0.567i)3-s + (0.0779 − 0.541i)4-s + (−0.416 − 0.190i)5-s + (−0.100 − 0.698i)6-s + (−0.415 + 1.41i)7-s + (−0.875 + 0.562i)8-s + (0.0417 + 0.0913i)9-s + (0.0867 + 0.295i)10-s + (−0.552 − 0.478i)11-s + (0.376 − 0.434i)12-s + (1.11 − 0.327i)13-s + (0.901 − 0.411i)14-s + (−0.259 − 0.404i)15-s + (0.146 + 0.0430i)16-s + (−0.774 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.841690 - 0.177475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.841690 - 0.177475i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-22.1 + 6.20i)T \) |
| good | 2 | \( 1 + (0.881 + 1.01i)T + (-0.569 + 3.95i)T^{2} \) |
| 3 | \( 1 + (-2.64 - 1.70i)T + (3.73 + 8.18i)T^{2} \) |
| 5 | \( 1 + (2.08 + 0.951i)T + (16.3 + 18.8i)T^{2} \) |
| 7 | \( 1 + (2.90 - 9.89i)T + (-41.2 - 26.4i)T^{2} \) |
| 11 | \( 1 + (6.07 + 5.26i)T + (17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-14.4 + 4.25i)T + (142. - 91.3i)T^{2} \) |
| 17 | \( 1 + (13.1 - 1.89i)T + (277. - 81.4i)T^{2} \) |
| 19 | \( 1 + (-26.2 - 3.77i)T + (346. + 101. i)T^{2} \) |
| 29 | \( 1 + (-0.753 - 5.23i)T + (-806. + 236. i)T^{2} \) |
| 31 | \( 1 + (29.7 - 19.1i)T + (399. - 874. i)T^{2} \) |
| 37 | \( 1 + (32.5 - 14.8i)T + (896. - 1.03e3i)T^{2} \) |
| 41 | \( 1 + (-7.60 + 16.6i)T + (-1.10e3 - 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-8.28 + 12.8i)T + (-768. - 1.68e3i)T^{2} \) |
| 47 | \( 1 - 72.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + (25.8 - 88.1i)T + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (71.1 - 20.8i)T + (2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (15.2 + 23.7i)T + (-1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (5.29 - 4.58i)T + (638. - 4.44e3i)T^{2} \) |
| 71 | \( 1 + (-18.7 - 21.6i)T + (-717. + 4.98e3i)T^{2} \) |
| 73 | \( 1 + (1.65 - 11.5i)T + (-5.11e3 - 1.50e3i)T^{2} \) |
| 79 | \( 1 + (-5.91 - 20.1i)T + (-5.25e3 + 3.37e3i)T^{2} \) |
| 83 | \( 1 + (15.0 - 6.85i)T + (4.51e3 - 5.20e3i)T^{2} \) |
| 89 | \( 1 + (-21.3 + 33.1i)T + (-3.29e3 - 7.20e3i)T^{2} \) |
| 97 | \( 1 + (77.7 + 35.4i)T + (6.16e3 + 7.11e3i)T^{2} \) |
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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
−18.14287690664937343191931716911, −15.66643909892620264905496319087, −15.50706724580438379338418176615, −13.95831138836016025728756514535, −12.17367779977486827364746689328, −10.79033567244261862943439898790, −9.231891547025923017877981442687, −8.606221946965487034964007726030, −5.74137923051203711925457380547, −2.99897238130722074075805631366,
3.50959259140113260326206463056, 7.07885428875466974222132406428, 7.71460660646040371062899645238, 9.193506155720072687808625249177, 11.12369738934736905519692447788, 13.04023352838451977902926988399, 13.76026400007671495055827154018, 15.47852907258902155995284640000, 16.49341297516718847727031548066, 17.75136145122288937999373352113
