| L(s) = 1 | + (−0.186 − 0.0949i)2-s + (1.71 + 0.270i)3-s + (−2.32 − 3.20i)4-s + (−1.31 − 4.82i)5-s + (−0.292 − 0.212i)6-s + (5.11 − 5.11i)7-s + (0.260 + 1.64i)8-s + (2.85 + 0.927i)9-s + (−0.212 + 1.02i)10-s + (3.21 + 9.89i)11-s + (−3.11 − 6.10i)12-s + (3.78 − 1.92i)13-s + (−1.43 + 0.467i)14-s + (−0.949 − 8.60i)15-s + (−4.78 + 14.7i)16-s + (−11.4 + 1.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.0931 − 0.0474i)2-s + (0.570 + 0.0903i)3-s + (−0.581 − 0.800i)4-s + (−0.263 − 0.964i)5-s + (−0.0488 − 0.0354i)6-s + (0.730 − 0.730i)7-s + (0.0325 + 0.205i)8-s + (0.317 + 0.103i)9-s + (−0.0212 + 0.102i)10-s + (0.292 + 0.899i)11-s + (−0.259 − 0.508i)12-s + (0.291 − 0.148i)13-s + (−0.102 + 0.0333i)14-s + (−0.0632 − 0.573i)15-s + (−0.298 + 0.920i)16-s + (−0.673 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07665 - 0.683481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07665 - 0.683481i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 - 0.270i)T \) |
| 5 | \( 1 + (1.31 + 4.82i)T \) |
| good | 2 | \( 1 + (0.186 + 0.0949i)T + (2.35 + 3.23i)T^{2} \) |
| 7 | \( 1 + (-5.11 + 5.11i)T - 49iT^{2} \) |
| 11 | \( 1 + (-3.21 - 9.89i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-3.78 + 1.92i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (11.4 - 1.81i)T + (274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-18.4 + 25.4i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (16.0 - 31.4i)T + (-310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-21.7 - 29.9i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-33.9 - 24.6i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (7.73 + 15.1i)T + (-804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-10.4 + 32.0i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (4.34 + 4.34i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-7.65 + 48.3i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-20.9 - 3.32i)T + (2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (31.2 + 10.1i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-28.9 - 89.0i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (122. - 19.3i)T + (4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-24.0 + 17.4i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-17.1 + 33.6i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (5.94 + 8.17i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (10.7 + 67.8i)T + (-6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (86.3 - 28.0i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (13.5 - 85.2i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
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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
−13.91354692691793695447116157405, −13.40104258440865092201563048496, −11.91983768399335190369957164860, −10.57985614725652961594433166340, −9.423237064160042237066182294383, −8.585246994311254165543057311969, −7.25472454582972863193872238136, −5.14146891648043528018179781547, −4.23140665982262400318311640216, −1.33838761855296703053546469779,
2.79275340284149993610176549620, 4.19312299970740479883245216187, 6.27994617627242077039760486857, 7.927291670019961255215638880506, 8.449363931700458395552849376419, 9.848363430368325872947122346278, 11.39511397709659921766262347290, 12.20479551244662203551713857163, 13.74417519405202052644255813949, 14.25588006824901896100210469901
