| L(s) = 1 | + (−0.112 − 0.782i)2-s + (0.909 + 0.415i)3-s + (1.31 − 0.387i)4-s + (1.72 − 1.99i)5-s + (0.222 − 0.758i)6-s + (2.48 − 0.913i)7-s + (−1.10 − 2.42i)8-s + (0.654 + 0.755i)9-s + (−1.75 − 1.12i)10-s + (2.41 + 0.347i)11-s + (1.36 + 0.195i)12-s + (−3.57 + 5.55i)13-s + (−0.994 − 1.83i)14-s + (2.40 − 1.09i)15-s + (0.539 − 0.346i)16-s + (−4.61 − 1.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.0795 − 0.553i)2-s + (0.525 + 0.239i)3-s + (0.659 − 0.193i)4-s + (0.773 − 0.892i)5-s + (0.0909 − 0.309i)6-s + (0.938 − 0.345i)7-s + (−0.391 − 0.858i)8-s + (0.218 + 0.251i)9-s + (−0.555 − 0.356i)10-s + (0.727 + 0.104i)11-s + (0.392 + 0.0564i)12-s + (−0.990 + 1.54i)13-s + (−0.265 − 0.491i)14-s + (0.620 − 0.283i)15-s + (0.134 − 0.0866i)16-s + (−1.11 − 0.328i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89120 - 1.13898i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89120 - 1.13898i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.48 + 0.913i)T \) |
| 23 | \( 1 + (3.22 - 3.55i)T \) |
| good | 2 | \( 1 + (0.112 + 0.782i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-1.72 + 1.99i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.41 - 0.347i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.57 - 5.55i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.61 + 1.35i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (8.15 - 2.39i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-5.41 - 1.59i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.351 + 0.160i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (3.16 - 2.73i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.75 - 3.25i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-7.91 - 3.61i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 + (4.81 + 7.49i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (0.711 - 1.10i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (2.14 + 4.70i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (5.63 - 0.810i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.314 + 2.18i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.31 - 7.87i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.38 + 5.26i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.37 - 2.73i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.867 + 1.90i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.63 + 4.19i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.82260335585689646268870196132, −9.869014929744464567127435559327, −9.209440361514259902501666418494, −8.421563373068707628508957030795, −7.09937006434311026654424767459, −6.28074002114234770947314986812, −4.79080429553735562262796992391, −4.09927019998017859226238107719, −2.17039198981748715259861226335, −1.67285165758393643778554905843,
2.21032039131225381663462670761, 2.64107572656506114827653383829, 4.47575251106239184805938878314, 5.95469396235116305383796010582, 6.49849344962449935343350964299, 7.49118327561193999098572036313, 8.295835775465058460280370791337, 9.067211025245932889480477852688, 10.67279273667802811748167512522, 10.68308844348508329994231661190
