| L(s) = 1 | + (0.347 − 1.96i)2-s + (−0.267 − 1.51i)3-s + (−3.75 − 1.36i)4-s + (−13.2 + 11.1i)5-s − 3.07·6-s + (−24.3 + 20.3i)7-s + (−4 + 6.92i)8-s + (23.1 − 8.42i)9-s + (17.3 + 30.0i)10-s + (13.5 − 23.4i)11-s + (−1.06 + 6.06i)12-s + (−45.8 − 16.6i)13-s + (31.7 + 54.9i)14-s + (20.4 + 17.1i)15-s + (12.2 + 10.2i)16-s + (−78.3 + 28.5i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.0514 − 0.291i)3-s + (−0.469 − 0.171i)4-s + (−1.18 + 0.995i)5-s − 0.209·6-s + (−1.31 + 1.10i)7-s + (−0.176 + 0.306i)8-s + (0.857 − 0.311i)9-s + (0.547 + 0.948i)10-s + (0.370 − 0.641i)11-s + (−0.0257 + 0.145i)12-s + (−0.977 − 0.355i)13-s + (0.605 + 1.04i)14-s + (0.351 + 0.295i)15-s + (0.191 + 0.160i)16-s + (−1.11 + 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.445 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.151391 + 0.244372i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151391 + 0.244372i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 + 1.96i)T \) |
| 37 | \( 1 + (-5.93 + 224. i)T \) |
| good | 3 | \( 1 + (0.267 + 1.51i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (13.2 - 11.1i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (24.3 - 20.3i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (-13.5 + 23.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (45.8 + 16.6i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (78.3 - 28.5i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 19 | \( 1 + (-8.75 - 49.6i)T + (-6.44e3 + 2.34e3i)T^{2} \) |
| 23 | \( 1 + (26.8 + 46.5i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (98.9 - 171. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 30.3T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-179. - 65.2i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + 125.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-297. - 514. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (165. + 139. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-626. - 525. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (507. + 184. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-108. + 91.2i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-95.6 - 542. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + 1.21e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (617. - 518. i)T + (8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (29.6 - 10.8i)T + (4.38e5 - 3.67e5i)T^{2} \) |
| 89 | \( 1 + (429. + 360. i)T + (1.22e5 + 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-202. - 350. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.51169124896589706947545117006, −12.84498527409843978718610496393, −12.33359090846192534166530051553, −11.27588518867355689303737970797, −10.10757807447229648308348960810, −8.905640914095748595848592519701, −7.30273153918487177669699014254, −6.15127109216672412891933237790, −3.93440980711648718709390694399, −2.75116923872722524198720279999,
0.17427927981030702378471412026, 4.01563570520960029415674613346, 4.66065807483593927676776403723, 6.88729714759067562771946135208, 7.54194642792380044703825877737, 9.168449456485088521739730540018, 10.03630142900099922927188714809, 11.75508740559942903844400288287, 12.86813101561667628416686956782, 13.53579061348822599069333563175
