| L(s) = 1 | + (0.951 − 0.309i)2-s + (−1.50 + 0.852i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (−1.17 + 1.27i)6-s + (−0.0129 − 0.123i)7-s + (0.587 − 0.809i)8-s + (1.54 − 2.57i)9-s + (−0.669 + 0.743i)10-s + (1.53 + 0.684i)11-s + (−0.718 + 1.57i)12-s + (0.248 + 1.17i)13-s + (−0.0505 − 0.113i)14-s + (0.879 − 1.49i)15-s + (0.309 − 0.951i)16-s + (0.251 − 0.111i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (−0.870 + 0.492i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (−0.477 + 0.521i)6-s + (−0.00490 − 0.0467i)7-s + (0.207 − 0.286i)8-s + (0.514 − 0.857i)9-s + (−0.211 + 0.235i)10-s + (0.463 + 0.206i)11-s + (−0.207 + 0.454i)12-s + (0.0690 + 0.324i)13-s + (−0.0135 − 0.0303i)14-s + (0.226 − 0.385i)15-s + (0.0772 − 0.237i)16-s + (0.0609 − 0.0271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67115 + 0.431600i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67115 + 0.431600i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (1.50 - 0.852i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (-3.31 - 4.46i)T \) |
| good | 7 | \( 1 + (0.0129 + 0.123i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-1.53 - 0.684i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.248 - 1.17i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.251 + 0.111i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-4.89 - 1.04i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (2.82 + 2.05i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.71 - 8.36i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-8.89 - 5.13i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.07 - 3.67i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.901 + 4.24i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (2.81 + 0.915i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.920 + 8.75i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-0.903 + 0.813i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 3.06iT - 61T^{2} \) |
| 67 | \( 1 + (-3.27 - 5.67i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (15.0 + 1.58i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (2.31 - 5.20i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-3.84 - 8.64i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (3.69 - 4.10i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-7.19 + 5.22i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.57 + 2.59i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.26040824036699163332491609726, −9.650246690640955947976772742638, −8.524977805182573888977396147737, −7.23351581722788173910918063213, −6.59670993398175717684487482104, −5.66949548313577553153397426628, −4.76035894156598606903321073186, −3.99542221937743190023482287265, −3.00731156679469451910092599246, −1.20941796782765110320254860562,
0.907514890447558770360273087934, 2.51834555534206808053002799256, 3.93968518632634710386640679289, 4.74734555072762573879432286727, 5.84581923180182513745165466642, 6.24112069700166127820481707640, 7.59770474566308225902511645962, 7.76245340929005322472676132875, 9.174997313555849438071683373797, 10.16793151768096879760699646489
