| L(s) = 1 | + (0.0475 + 0.0305i)2-s + (0.989 − 0.142i)3-s + (−0.829 − 1.81i)4-s + (−1.34 − 0.396i)5-s + (0.0514 + 0.0235i)6-s + (1.78 − 1.94i)7-s + (0.0321 − 0.223i)8-s + (0.959 − 0.281i)9-s + (−0.0521 − 0.0601i)10-s + (−2.64 − 4.12i)11-s + (−1.07 − 1.67i)12-s + (−5.25 + 4.55i)13-s + (0.144 − 0.0380i)14-s + (−1.39 − 0.200i)15-s + (−2.60 + 3.00i)16-s + (2.10 − 4.60i)17-s + ⋯ |
| L(s) = 1 | + (0.0336 + 0.0216i)2-s + (0.571 − 0.0821i)3-s + (−0.414 − 0.908i)4-s + (−0.603 − 0.177i)5-s + (0.0210 + 0.00959i)6-s + (0.676 − 0.736i)7-s + (0.0113 − 0.0791i)8-s + (0.319 − 0.0939i)9-s + (−0.0164 − 0.0190i)10-s + (−0.798 − 1.24i)11-s + (−0.311 − 0.484i)12-s + (−1.45 + 1.26i)13-s + (0.0386 − 0.0101i)14-s + (−0.359 − 0.0516i)15-s + (−0.651 + 0.752i)16-s + (0.510 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.578052 - 1.02695i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.578052 - 1.02695i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.78 + 1.94i)T \) |
| 23 | \( 1 + (2.49 + 4.09i)T \) |
| good | 2 | \( 1 + (-0.0475 - 0.0305i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (1.34 + 0.396i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (2.64 + 4.12i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (5.25 - 4.55i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.10 + 4.60i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.25 - 4.94i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.06 + 4.52i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-6.92 - 0.995i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.69 + 5.77i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.55 + 5.29i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-4.87 + 0.700i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 1.42iT - 47T^{2} \) |
| 53 | \( 1 + (-9.48 - 8.21i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (4.73 - 4.10i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.654 - 4.55i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.14 + 3.34i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-4.34 - 2.79i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.71 + 1.23i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.80 - 3.29i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.35 + 0.690i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (2.06 + 14.3i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-7.20 - 2.11i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.47323962607716795670450903867, −9.886123926359254178018586095101, −8.863038439862146001521807425320, −7.939702972963313718515746604037, −7.27084230969123931908229003363, −5.88320958286265728574447263784, −4.76961663094802107362428729896, −4.03742111303235788694509192879, −2.37984932484903624582628041261, −0.66393092739094258118078832432,
2.37073333864018006657788904196, 3.25367506539227680553030639069, 4.60195541670477258146245689543, 5.24929625674612537268876016612, 7.19419834772167586499214514583, 7.87387381317327153486032496429, 8.241394815245992964878901780169, 9.509569286180352547748803294146, 10.18981103590934308185156307786, 11.50154423557905763444720536328
