| L(s) = 1 | + (−1.87 − 4.77i)2-s + (−0.361 − 4.82i)3-s + (−13.3 + 12.4i)4-s + (−6.84 + 4.66i)5-s + (−22.3 + 10.7i)6-s + (8.66 − 16.3i)7-s + (47.3 + 22.8i)8-s + (3.55 − 0.535i)9-s + (35.0 + 23.9i)10-s + (−48.9 − 7.37i)11-s + (64.7 + 60.0i)12-s + (−5.85 + 7.34i)13-s + (−94.3 − 10.6i)14-s + (24.9 + 31.3i)15-s + (9.22 − 123. i)16-s + (−80.6 + 24.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.661 − 1.68i)2-s + (−0.0695 − 0.928i)3-s + (−1.67 + 1.55i)4-s + (−0.611 + 0.417i)5-s + (−1.51 + 0.731i)6-s + (0.467 − 0.883i)7-s + (2.09 + 1.00i)8-s + (0.131 − 0.0198i)9-s + (1.10 + 0.755i)10-s + (−1.34 − 0.202i)11-s + (1.55 + 1.44i)12-s + (−0.124 + 0.156i)13-s + (−1.80 − 0.203i)14-s + (0.429 + 0.539i)15-s + (0.144 − 1.92i)16-s + (−1.15 + 0.354i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.283379 + 0.461241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.283379 + 0.461241i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-8.66 + 16.3i)T \) |
| good | 2 | \( 1 + (1.87 + 4.77i)T + (-5.86 + 5.44i)T^{2} \) |
| 3 | \( 1 + (0.361 + 4.82i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (6.84 - 4.66i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (48.9 + 7.37i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (5.85 - 7.34i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (80.6 - 24.8i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-59.8 + 103. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (23.3 + 7.19i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-51.4 + 225. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (103. + 178. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-263. - 244. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (65.4 + 31.4i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-279. + 134. i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (13.8 + 35.1i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-286. + 265. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-315. - 215. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (258. + 239. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-328. - 569. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-133. - 586. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-101. + 258. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-14.3 + 24.8i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (330. + 414. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (1.13e3 - 170. i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 824.T + 9.12e5T^{2} \) |
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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
−13.44720569924512356325660502626, −13.14917269896092924992210852043, −11.63424303477067078065692631172, −11.04604728731363453073728806903, −9.883401608273252467585204631896, −8.172371788751137617956652292164, −7.29004256262415544353385355822, −4.23736226099557181616847425813, −2.39782372724691874405901380371, −0.52240080394910227366838007736,
4.65643374910172233294327930075, 5.55091786904164167141772379117, 7.45388589646169947717801107088, 8.441841129412185589252739561106, 9.484914755008572097536905763680, 10.71659172952858981173250555540, 12.60232415110542591736412382079, 14.29348124141706896427195875952, 15.32278114993088768886205966394, 15.92667184909445747127832633937
