| 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
−11.52366501137912458678190595312, −10.10242980551391251073457372594, −9.463161120520936481255388878950, −8.329328717707913125228417228982, −7.70198340008315267830243416995, −6.25258934943806942885724591420, −5.33112304578452952009246842847, −4.58025477823976017855697544475, −3.33104760062179581985803527530, −1.11721827573253598817623445816,
0.45798572770924812961893823497, 3.24740518078396730769011558600, 3.98186721122768138164594149044, 4.78601877082027074403421833664, 6.67814037008102329022883349385, 7.20449675302681071654835060614, 8.040899504012482457638639101421, 8.763234717694189693662166560921, 10.26248925014249509406741077689, 10.97616756431516726472235730675
