| L(s) = 1 | + (0.109 + 0.0793i)2-s + (0.927 + 2.85i)3-s + (−2.46 − 7.59i)4-s + (−6.22 − 9.28i)5-s + (−0.125 + 0.385i)6-s − 17.3·7-s + (0.666 − 2.05i)8-s + (−7.28 + 5.29i)9-s + (0.0573 − 1.50i)10-s + (−34.1 − 24.7i)11-s + (19.3 − 14.0i)12-s + (68.3 − 49.6i)13-s + (−1.89 − 1.37i)14-s + (20.7 − 26.3i)15-s + (−51.4 + 37.3i)16-s + (−13.9 + 42.8i)17-s + ⋯ |
| L(s) = 1 | + (0.0386 + 0.0280i)2-s + (0.178 + 0.549i)3-s + (−0.308 − 0.948i)4-s + (−0.556 − 0.830i)5-s + (−0.00851 + 0.0262i)6-s − 0.934·7-s + (0.0294 − 0.0906i)8-s + (−0.269 + 0.195i)9-s + (0.00181 − 0.0476i)10-s + (−0.934 − 0.679i)11-s + (0.466 − 0.338i)12-s + (1.45 − 1.05i)13-s + (−0.0360 − 0.0262i)14-s + (0.356 − 0.453i)15-s + (−0.803 + 0.583i)16-s + (−0.198 + 0.611i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.429515 - 0.737306i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.429515 - 0.737306i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.927 - 2.85i)T \) |
| 5 | \( 1 + (6.22 + 9.28i)T \) |
| good | 2 | \( 1 + (-0.109 - 0.0793i)T + (2.47 + 7.60i)T^{2} \) |
| 7 | \( 1 + 17.3T + 343T^{2} \) |
| 11 | \( 1 + (34.1 + 24.7i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-68.3 + 49.6i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (13.9 - 42.8i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-25.8 + 79.5i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (16.9 + 12.3i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-68.0 - 209. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-49.5 + 152. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-212. + 154. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (331. - 241. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (142. + 438. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (44.0 + 135. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-382. + 277. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (43.7 + 31.7i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-139. + 427. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (193. + 595. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (39.4 + 28.6i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (200. + 615. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-206. + 637. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-700. - 509. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (54.7 + 168. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−13.40687638930832964135230103769, −13.04032394220556075776197195022, −11.17311040265698884554715461389, −10.32588283089071907026878991760, −9.109959588945285320004350965689, −8.223382581868974236536689030542, −6.12738787062868121916390783850, −5.04072718762182864301282933958, −3.47231582852086846306199580522, −0.52962641031774565208289444171,
2.79282821457685224554808780956, 4.02454250194091312590327069913, 6.40502448446631637706003984093, 7.42269353390679690128178304548, 8.448442682194050707448730757062, 9.869975518011324552515089915316, 11.37780947548168439572934816443, 12.28169404329182035475979971351, 13.33414589476481676699368344868, 14.06888842884162304659238952512
