| 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 + (−0.195 + 1.86i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)10-s + (5.36 − 2.39i)11-s + (−0.978 + 0.207i)12-s + (1.33 + 0.283i)13-s + (1.71 + 0.762i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−6.88 − 3.06i)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 + (−0.0740 + 0.704i)7-s + (−0.286 + 0.207i)8-s + (−0.0348 − 0.331i)9-s + (−0.211 − 0.235i)10-s + (1.61 − 0.720i)11-s + (−0.282 + 0.0600i)12-s + (0.370 + 0.0786i)13-s + (0.457 + 0.203i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−1.66 − 0.743i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17205 - 1.71625i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17205 - 1.71625i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (5.35 - 1.52i)T \) |
| good | 7 | \( 1 + (0.195 - 1.86i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-5.36 + 2.39i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.33 - 0.283i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (6.88 + 3.06i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-6.14 + 1.30i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.26 + 1.64i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.22 + 3.75i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.186 + 0.323i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.64 + 4.05i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (0.499 - 0.106i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (1.78 + 5.49i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.35 - 12.8i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-4.34 + 4.82i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 6.57T + 61T^{2} \) |
| 67 | \( 1 + (4.64 - 8.03i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.543 - 5.16i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-4.15 + 1.85i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (9.27 + 4.13i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-9.74 - 10.8i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-7.65 - 5.55i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-8.03 - 5.83i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.493557746711766610136706340765, −9.020760966632745138446350976478, −8.607619206440676830345945244060, −7.15284720456582627276479048470, −6.32722783070115574827136292700, −5.40058957617284744123270424128, −4.27359000054376146386507116521, −3.25066784939795000351671500399, −2.17922813291680362061844909552, −0.974272404381085588304838325935,
1.64634536697223679807423170898, 3.37498715980473351280969586742, 4.04638423065227134998652529101, 4.96725338404535481756728787754, 6.24253560390659863021267409899, 6.88827335459012830409335628850, 7.61289465461895829159392715665, 8.809046650956721096038733024067, 9.313158366128640236078046852350, 10.15705163353702786026194952141
