| L(s) = 1 | + (0.819 − 0.573i)2-s + (1.71 + 0.236i)3-s + (0.342 − 0.939i)4-s + (0.250 − 2.22i)5-s + (1.54 − 0.790i)6-s + (0.373 + 0.100i)7-s + (−0.258 − 0.965i)8-s + (2.88 + 0.810i)9-s + (−1.06 − 1.96i)10-s + (−3.61 − 2.08i)11-s + (0.808 − 1.53i)12-s + (0.240 + 0.0210i)13-s + (0.363 − 0.132i)14-s + (0.954 − 3.75i)15-s + (−0.766 − 0.642i)16-s + (−0.0718 + 0.0503i)17-s + ⋯ |
| L(s) = 1 | + (0.579 − 0.405i)2-s + (0.990 + 0.136i)3-s + (0.171 − 0.469i)4-s + (0.112 − 0.993i)5-s + (0.629 − 0.322i)6-s + (0.141 + 0.0378i)7-s + (−0.0915 − 0.341i)8-s + (0.962 + 0.270i)9-s + (−0.338 − 0.621i)10-s + (−1.09 − 0.629i)11-s + (0.233 − 0.442i)12-s + (0.0667 + 0.00583i)13-s + (0.0971 − 0.0353i)14-s + (0.246 − 0.969i)15-s + (−0.191 − 0.160i)16-s + (−0.0174 + 0.0122i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.29118 - 1.51544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29118 - 1.51544i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.819 + 0.573i)T \) |
| 3 | \( 1 + (-1.71 - 0.236i)T \) |
| 5 | \( 1 + (-0.250 + 2.22i)T \) |
| 19 | \( 1 + (-3.07 - 3.08i)T \) |
| good | 7 | \( 1 + (-0.373 - 0.100i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (3.61 + 2.08i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.240 - 0.0210i)T + (12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (0.0718 - 0.0503i)T + (5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (-1.28 + 0.598i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.577 - 3.27i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.14 - 3.71i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.64 - 7.64i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.48 + 4.15i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.61 - 1.22i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (0.0805 - 0.115i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-6.98 + 3.25i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (0.127 - 0.724i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.62 - 2.40i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.994 - 0.696i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (-1.77 - 4.86i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.355 - 4.05i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (9.94 - 11.8i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (8.49 + 2.27i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.19 + 1.00i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (6.33 + 9.04i)T + (-33.1 + 91.1i)T^{2} \) |
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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
−10.41294557022657083032513862320, −9.817693982851412776389938573113, −8.684739546268924234076519336117, −8.222529893798289379324635265906, −7.07487317205415122932069786098, −5.57495135247799982371779803765, −4.89700054166583617248725899415, −3.75213085774720333500723004928, −2.73511840452037836533558036330, −1.38343378532801051672696728948,
2.24422685387232336810465244517, 3.03158600879697914643147961759, 4.16231933617084567911748389298, 5.32755855563407182529320826920, 6.56036768909542798432378898637, 7.43351709029480879810373827695, 7.85500423498623891858874881256, 9.100424086465592287729140914163, 10.00304779803493235677941790497, 10.83547752893196265865890232635
