| 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 + (−0.540 − 5.13i)7-s + (0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)10-s + (−1.96 − 0.875i)11-s + (−0.978 − 0.207i)12-s + (−5.94 + 1.26i)13-s + (−4.72 + 2.10i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−5.35 + 2.38i)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 + (−0.204 − 1.94i)7-s + (0.286 + 0.207i)8-s + (−0.0348 + 0.331i)9-s + (0.211 − 0.235i)10-s + (−0.593 − 0.264i)11-s + (−0.282 − 0.0600i)12-s + (−1.64 + 0.350i)13-s + (−1.26 + 0.561i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (−1.29 + 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0203352 + 0.404354i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0203352 + 0.404354i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.79 + 4.07i)T \) |
| good | 7 | \( 1 + (0.540 + 5.13i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (1.96 + 0.875i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (5.94 - 1.26i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (5.35 - 2.38i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (1.35 + 0.288i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.725 + 0.526i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.410 + 1.26i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.21 - 3.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.43 + 8.25i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (3.44 + 0.731i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (1.54 - 4.76i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.922 - 8.77i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (0.484 + 0.538i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 2.66T + 61T^{2} \) |
| 67 | \( 1 + (7.33 + 12.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.0601 + 0.571i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (8.43 + 3.75i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-1.31 + 0.585i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-9.27 + 10.2i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-8.39 + 6.10i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.37 + 1.72i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.829326808213745724661356521916, −9.098849278314172981139123613533, −7.87302144463564751584775046917, −7.34406342232002297898010610672, −6.34977960264243733227623737791, −4.68207294070411690945405115119, −4.23315277844716478347175198569, −3.09648716878401723013915302130, −2.05335264822663965543034733899, −0.17659408588871997983649194935,
2.13038387190161740667789839772, 2.77674731343841406162049537883, 4.74577550697605871211408847203, 5.31703663678300292171953390486, 6.26941236038987589622849419968, 7.13520575787076364123150946311, 8.106867806030817076450042542362, 8.764201854854479171285515296537, 9.407076872312351789775896471529, 10.07500234816899320956533581691
