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
−10.83547752893196265865890232635, −10.00304779803493235677941790497, −9.100424086465592287729140914163, −7.85500423498623891858874881256, −7.43351709029480879810373827695, −6.56036768909542798432378898637, −5.32755855563407182529320826920, −4.16231933617084567911748389298, −3.03158600879697914643147961759, −2.24422685387232336810465244517,
1.38343378532801051672696728948, 2.73511840452037836533558036330, 3.75213085774720333500723004928, 4.89700054166583617248725899415, 5.57495135247799982371779803765, 7.07487317205415122932069786098, 8.222529893798289379324635265906, 8.684739546268924234076519336117, 9.817693982851412776389938573113, 10.41294557022657083032513862320