| L(s) = 1 | + (0.230 − 0.180i)2-s + (−0.814 + 0.580i)3-s + (−0.451 + 1.86i)4-s + (−1.27 − 0.245i)5-s + (−0.0824 + 0.280i)6-s + (2.61 + 0.430i)7-s + (0.475 + 1.04i)8-s + (0.327 − 0.945i)9-s + (−0.337 + 0.173i)10-s + (0.611 − 0.778i)11-s + (−0.711 − 1.77i)12-s + (−2.09 + 3.25i)13-s + (0.678 − 0.373i)14-s + (1.18 − 0.538i)15-s + (−3.10 − 1.60i)16-s + (−4.95 + 4.72i)17-s + ⋯ |
| L(s) = 1 | + (0.162 − 0.127i)2-s + (−0.470 + 0.334i)3-s + (−0.225 + 0.930i)4-s + (−0.569 − 0.109i)5-s + (−0.0336 + 0.114i)6-s + (0.986 + 0.162i)7-s + (0.168 + 0.368i)8-s + (0.109 − 0.315i)9-s + (−0.106 + 0.0550i)10-s + (0.184 − 0.234i)11-s + (−0.205 − 0.513i)12-s + (−0.580 + 0.903i)13-s + (0.181 − 0.0997i)14-s + (0.304 − 0.139i)15-s + (−0.776 − 0.400i)16-s + (−1.20 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.384991 + 0.814335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.384991 + 0.814335i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 - 0.580i)T \) |
| 7 | \( 1 + (-2.61 - 0.430i)T \) |
| 23 | \( 1 + (3.75 + 2.97i)T \) |
| good | 2 | \( 1 + (-0.230 + 0.180i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (1.27 + 0.245i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (-0.611 + 0.778i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.09 - 3.25i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.95 - 4.72i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.13 - 3.94i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (4.64 + 1.36i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.638 - 6.69i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (4.80 + 1.66i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-5.07 - 4.39i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.50 - 2.05i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-1.16 + 0.671i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-12.3 + 0.587i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (3.40 + 6.61i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-3.79 + 5.33i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (1.03 - 2.59i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.06 - 7.44i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-11.5 - 2.81i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.687 + 0.0327i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-5.99 - 6.91i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.895 + 0.0855i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (9.96 - 11.4i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.46804611766550752304527590844, −10.70931566117896716346920094268, −9.420890950187493244257342448540, −8.482054650213244905812881470507, −7.85560513218925977043050230978, −6.78847655347414754633518188341, −5.45495556764770627060000078458, −4.32839661576150941726152756098, −3.85640787041558095970822217490, −2.06884253999181975749454378745,
0.54814562739479007471951097028, 2.17295768581094303309655845416, 4.13168031066517763409789670086, 5.07118882279378484286176557430, 5.74399010159317580227204909468, 7.20352633507244924952659088825, 7.53751511183862103706981379551, 8.966161773927497516474570660077, 9.828862647898291481651176209753, 10.91434180442874675375398996709
