| L(s) = 1 | + (−2.07 − 1.33i)2-s + (−0.989 + 0.142i)3-s + (1.68 + 3.69i)4-s + (3.92 + 1.15i)5-s + (2.23 + 1.02i)6-s + (1.23 − 2.33i)7-s + (0.723 − 5.03i)8-s + (0.959 − 0.281i)9-s + (−6.59 − 7.60i)10-s + (1.89 + 2.94i)11-s + (−2.19 − 3.41i)12-s + (0.793 − 0.687i)13-s + (−5.67 + 3.19i)14-s + (−4.04 − 0.581i)15-s + (−2.87 + 3.31i)16-s + (−2.66 + 5.83i)17-s + ⋯ |
| L(s) = 1 | + (−1.46 − 0.941i)2-s + (−0.571 + 0.0821i)3-s + (0.844 + 1.84i)4-s + (1.75 + 0.515i)5-s + (0.914 + 0.417i)6-s + (0.467 − 0.883i)7-s + (0.255 − 1.77i)8-s + (0.319 − 0.0939i)9-s + (−2.08 − 2.40i)10-s + (0.570 + 0.887i)11-s + (−0.634 − 0.986i)12-s + (0.220 − 0.190i)13-s + (−1.51 + 0.854i)14-s + (−1.04 − 0.150i)15-s + (−0.718 + 0.828i)16-s + (−0.646 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.831975 - 0.160535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.831975 - 0.160535i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.23 + 2.33i)T \) |
| 23 | \( 1 + (-4.30 + 2.11i)T \) |
| good | 2 | \( 1 + (2.07 + 1.33i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (-3.92 - 1.15i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-1.89 - 2.94i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.793 + 0.687i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.66 - 5.83i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.346 - 0.758i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.92 - 6.39i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (8.75 + 1.25i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (0.804 + 2.73i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (1.15 - 3.93i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-9.48 + 1.36i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 7.61iT - 47T^{2} \) |
| 53 | \( 1 + (0.894 + 0.775i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.11 + 5.30i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.390 - 2.71i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-3.02 + 4.70i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-11.9 - 7.65i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.76 + 1.26i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (2.98 - 2.58i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (3.67 - 1.07i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.689 - 4.79i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (3.41 + 1.00i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.88912308336646047425370737122, −10.12046778140015467932719728699, −9.458420443690041593595676480603, −8.653860378220572834071152344948, −7.26472302485493327395164351626, −6.64663933392692361102282639178, −5.36292351244298306439827393758, −3.76754805242277959132796910447, −2.10773736632511567207375568079, −1.39491516087072130121615717329,
1.05419268375631643241050482105, 2.23007696878549126352443884982, 5.08465193888221728024296810030, 5.75166675280906006658080068917, 6.36203719571852608194573570178, 7.36604531379806537849116743172, 8.707179950493804822064559731275, 9.193037814954430853541642503328, 9.626659447993574616746948423428, 10.87247255269508639558821741022
