| L(s) = 1 | + (−0.129 + 0.0829i)2-s + (−0.989 − 0.142i)3-s + (−0.821 + 1.79i)4-s + (−3.74 + 1.09i)5-s + (0.139 − 0.0637i)6-s + (−0.158 − 2.64i)7-s + (−0.0867 − 0.603i)8-s + (0.959 + 0.281i)9-s + (0.391 − 0.452i)10-s + (0.170 − 0.265i)11-s + (1.06 − 1.66i)12-s + (1.14 + 0.995i)13-s + (0.239 + 0.327i)14-s + (3.86 − 0.555i)15-s + (−2.52 − 2.91i)16-s + (2.42 + 5.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.0912 + 0.0586i)2-s + (−0.571 − 0.0821i)3-s + (−0.410 + 0.898i)4-s + (−1.67 + 0.491i)5-s + (0.0569 − 0.0260i)6-s + (−0.0598 − 0.998i)7-s + (−0.0306 − 0.213i)8-s + (0.319 + 0.0939i)9-s + (0.123 − 0.142i)10-s + (0.0515 − 0.0801i)11-s + (0.308 − 0.479i)12-s + (0.318 + 0.276i)13-s + (0.0639 + 0.0875i)14-s + (0.996 − 0.143i)15-s + (−0.631 − 0.729i)16-s + (0.588 + 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.484118 - 0.218278i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.484118 - 0.218278i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.989 + 0.142i)T \) |
| 7 | \( 1 + (0.158 + 2.64i)T \) |
| 23 | \( 1 + (1.47 + 4.56i)T \) |
| good | 2 | \( 1 + (0.129 - 0.0829i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (3.74 - 1.09i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (-0.170 + 0.265i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.14 - 0.995i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.42 - 5.31i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-3.19 + 6.98i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.19 - 4.81i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-4.53 + 0.651i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.789 + 2.69i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.21 + 7.53i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (7.04 + 1.01i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 8.31iT - 47T^{2} \) |
| 53 | \( 1 + (0.237 - 0.205i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.835 - 0.724i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.591 - 4.11i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (7.91 + 12.3i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (1.38 - 0.890i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-8.12 - 3.71i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.66 - 3.17i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (5.94 + 1.74i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.08 + 7.52i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-18.1 + 5.34i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.97616756431516726472235730675, −10.26248925014249509406741077689, −8.763234717694189693662166560921, −8.040899504012482457638639101421, −7.20449675302681071654835060614, −6.67814037008102329022883349385, −4.78601877082027074403421833664, −3.98186721122768138164594149044, −3.24740518078396730769011558600, −0.45798572770924812961893823497,
1.11721827573253598817623445816, 3.33104760062179581985803527530, 4.58025477823976017855697544475, 5.33112304578452952009246842847, 6.25258934943806942885724591420, 7.70198340008315267830243416995, 8.329328717707913125228417228982, 9.463161120520936481255388878950, 10.10242980551391251073457372594, 11.52366501137912458678190595312
