L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 + 0.587i)4-s + (0.5 + 0.866i)5-s + (0.499 − 0.866i)6-s + (0.0785 + 0.747i)7-s + (0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)10-s + (−3.05 − 1.36i)11-s + (−0.978 − 0.207i)12-s + (0.0198 − 0.00422i)13-s + (0.686 − 0.305i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (2.17 − 0.969i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.386 + 0.429i)3-s + (−0.404 + 0.293i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (0.0296 + 0.282i)7-s + (0.286 + 0.207i)8-s + (−0.0348 + 0.331i)9-s + (0.211 − 0.235i)10-s + (−0.921 − 0.410i)11-s + (−0.282 − 0.0600i)12-s + (0.00551 − 0.00117i)13-s + (0.183 − 0.0817i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.528 − 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.796 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42142 + 0.478897i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42142 + 0.478897i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.00514 - 5.56i)T \) |
good | 7 | \( 1 + (-0.0785 - 0.747i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (3.05 + 1.36i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.0198 + 0.00422i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.17 + 0.969i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-7.99 - 1.69i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.30 - 2.39i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.89 - 5.82i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (3.16 - 5.48i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.87 - 7.63i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (4.19 + 0.890i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (1.44 - 4.43i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.517 - 4.92i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (9.39 + 10.4i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 - 7.78T + 61T^{2} \) |
| 67 | \( 1 + (-6.98 - 12.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.39 + 13.2i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-8.26 - 3.68i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (12.8 - 5.72i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-9.04 + 10.0i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-9.44 + 6.86i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.71 - 4.15i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.08669743615393534856184390953, −9.549028956615262293897603206419, −8.602988506976281093981391552361, −7.86087878495753338108968331329, −6.93081211308167712740130213450, −5.46722702677850223355340387471, −4.94243619695987778458832731410, −3.19535658822186995565795951636, −3.10227597081131832496421584119, −1.45738860370137925013994414749,
0.796480232474184699255210528142, 2.31275904280437634133170427978, 3.65589860621751127194720548962, 4.97215013208740419400682202213, 5.60262597586441032358216512260, 6.78554374902143819577913296026, 7.53436056353340423732977236549, 8.115047205526635522349909859020, 9.068311002644548692014151326294, 9.798148071554569990655495119538