| L(s) = 1 | + (0.0271 + 0.188i)2-s + (−0.909 − 0.415i)3-s + (1.88 − 0.553i)4-s + (1.98 − 2.29i)5-s + (0.0537 − 0.183i)6-s + (−2.47 − 0.941i)7-s + (0.314 + 0.688i)8-s + (0.654 + 0.755i)9-s + (0.487 + 0.313i)10-s + (3.07 + 0.442i)11-s + (−1.94 − 0.279i)12-s + (1.29 − 2.01i)13-s + (0.110 − 0.492i)14-s + (−2.76 + 1.26i)15-s + (3.18 − 2.04i)16-s + (−6.24 − 1.83i)17-s + ⋯ |
| L(s) = 1 | + (0.0192 + 0.133i)2-s + (−0.525 − 0.239i)3-s + (0.942 − 0.276i)4-s + (0.889 − 1.02i)5-s + (0.0219 − 0.0747i)6-s + (−0.934 − 0.356i)7-s + (0.111 + 0.243i)8-s + (0.218 + 0.251i)9-s + (0.154 + 0.0991i)10-s + (0.928 + 0.133i)11-s + (−0.561 − 0.0806i)12-s + (0.359 − 0.558i)13-s + (0.0296 − 0.131i)14-s + (−0.713 + 0.325i)15-s + (0.795 − 0.511i)16-s + (−1.51 − 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31536 - 0.899078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31536 - 0.899078i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (2.47 + 0.941i)T \) |
| 23 | \( 1 + (0.644 - 4.75i)T \) |
| good | 2 | \( 1 + (-0.0271 - 0.188i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-1.98 + 2.29i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-3.07 - 0.442i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.29 + 2.01i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (6.24 + 1.83i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (1.82 - 0.535i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (5.49 + 1.61i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-7.97 + 3.64i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.32 + 5.47i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.97 - 4.31i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.26 + 1.03i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 4.70iT - 47T^{2} \) |
| 53 | \( 1 + (4.94 + 7.69i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (1.62 - 2.53i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.0813 - 0.178i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-12.0 + 1.73i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.06 - 7.43i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.328 - 1.11i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (8.23 - 12.8i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-7.53 - 8.69i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (2.14 - 4.69i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (8.54 - 9.85i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.96110893881495109150370947041, −9.755165785310337967184658322397, −9.352785085663473347752858230035, −7.985933377664219036852645400146, −6.77994691739589287034856809581, −6.22419236926805026766448449787, −5.44581641404119465024918498638, −4.11768661709188617707682990695, −2.36067732090850834078786683273, −1.06737004499198991176978996200,
1.99879752950324561604286576953, 3.01928058862165298385537383127, 4.24016911352290941104067161913, 6.15935471147357953030037487820, 6.35932353988434769795096798404, 6.96519424397335979700039096595, 8.651500581471130855970887689307, 9.584048306605436064776868038081, 10.47488893217908499986591792355, 11.04012925269627203700388345889
