| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−1.68 + 0.393i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (1.48 − 0.895i)6-s + (0.503 + 4.79i)7-s + (−0.587 + 0.809i)8-s + (2.69 − 1.32i)9-s + (−0.669 + 0.743i)10-s + (3.34 + 1.48i)11-s + (−1.13 + 1.30i)12-s + (0.191 + 0.900i)13-s + (−1.96 − 4.40i)14-s + (−1.26 + 1.18i)15-s + (0.309 − 0.951i)16-s + (3.22 − 1.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.973 + 0.227i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (0.605 − 0.365i)6-s + (0.190 + 1.81i)7-s + (−0.207 + 0.286i)8-s + (0.896 − 0.442i)9-s + (−0.211 + 0.235i)10-s + (1.00 + 0.449i)11-s + (−0.327 + 0.378i)12-s + (0.0530 + 0.249i)13-s + (−0.523 − 1.17i)14-s + (−0.326 + 0.305i)15-s + (0.0772 − 0.237i)16-s + (0.782 − 0.348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.599291 + 0.697163i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.599291 + 0.697163i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.68 - 0.393i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (-4.68 + 3.00i)T \) |
| good | 7 | \( 1 + (-0.503 - 4.79i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-3.34 - 1.48i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.191 - 0.900i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.22 + 1.43i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (2.91 + 0.618i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.0893 + 0.0649i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.02 + 3.14i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.546 + 0.315i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.90 - 7.12i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (2.30 - 10.8i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-1.38 - 0.451i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.987 + 9.39i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-0.439 + 0.395i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 6.64iT - 61T^{2} \) |
| 67 | \( 1 + (-6.53 - 11.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (11.4 + 1.19i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.190 - 0.426i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-1.63 - 3.66i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (6.02 - 6.69i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-8.73 + 6.34i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (9.20 - 6.68i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.971555546497153213231361697955, −9.535041108139086737214380177190, −8.835475595565984955358844475709, −7.88856721863778705954532631715, −6.60400741836737851642833251050, −6.07863059682691901061003007331, −5.32097161623027085562084285984, −4.33887620667953706885412077160, −2.54581495082104696631349573736, −1.34525405177197914659568807321,
0.71164504744202728854481120444, 1.62678675232267869409125092101, 3.53272016058603687456849925350, 4.37747463732786275358008542164, 5.71251959406891079842790847770, 6.62290377419762936455874506395, 7.19713512929328291710710991002, 8.016145150219845149162248820974, 9.158274949569599844542111382774, 10.27797219747060831691231414050
