| L(s) = 1 | + (−0.643 + 1.85i)2-s + (−0.458 − 0.888i)3-s + (−1.47 − 1.15i)4-s + (2.63 − 0.251i)5-s + (1.94 − 0.280i)6-s + (−0.893 − 2.49i)7-s + (−0.208 + 0.134i)8-s + (−0.580 + 0.814i)9-s + (−1.22 + 5.06i)10-s + (0.479 − 0.165i)11-s + (−0.354 + 1.84i)12-s + (−0.303 − 1.03i)13-s + (5.20 − 0.0585i)14-s + (−1.43 − 2.22i)15-s + (−0.998 − 4.11i)16-s + (4.52 + 1.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.455 + 1.31i)2-s + (−0.264 − 0.513i)3-s + (−0.736 − 0.579i)4-s + (1.17 − 0.112i)5-s + (0.795 − 0.114i)6-s + (−0.337 − 0.941i)7-s + (−0.0737 + 0.0474i)8-s + (−0.193 + 0.271i)9-s + (−0.388 + 1.60i)10-s + (0.144 − 0.0500i)11-s + (−0.102 + 0.531i)12-s + (−0.0840 − 0.286i)13-s + (1.39 − 0.0156i)14-s + (−0.369 − 0.575i)15-s + (−0.249 − 1.02i)16-s + (1.09 + 0.439i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.15754 + 0.334726i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15754 + 0.334726i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.458 + 0.888i)T \) |
| 7 | \( 1 + (0.893 + 2.49i)T \) |
| 23 | \( 1 + (-3.71 + 3.03i)T \) |
| good | 2 | \( 1 + (0.643 - 1.85i)T + (-1.57 - 1.23i)T^{2} \) |
| 5 | \( 1 + (-2.63 + 0.251i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (-0.479 + 0.165i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (0.303 + 1.03i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-4.52 - 1.81i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-6.47 + 2.59i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (1.13 + 7.91i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (4.86 - 0.231i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-7.84 - 5.58i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (-8.01 - 3.66i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (6.39 - 9.95i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-3.75 + 2.16i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.84 - 4.03i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (4.26 + 1.03i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (9.66 + 4.98i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (1.74 + 9.04i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (5.93 + 6.84i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (5.40 - 6.87i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-10.8 + 11.4i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-1.83 - 4.02i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.537 - 11.2i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (-0.334 + 0.731i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.88647791314432688565048866206, −9.725411319962919352877685917419, −9.378804751949732775544304325359, −7.930313775023794576320786733966, −7.46930022798652329185079621639, −6.38123610505269932142540206123, −5.90238097557666209074961486410, −4.86599443230774340726141035482, −2.94406444877052444796281459354, −1.02947271711789049170462686650,
1.43440178404198826764159209453, 2.68309779005558241417919215232, 3.57907730693970460422728818052, 5.39096603663382513441619459407, 5.84269806292683707646491566336, 7.27982069719541997373135037483, 9.007542659325732379865513746285, 9.351970517215952746613221931051, 9.974962814517489382132942684168, 10.78127792335115923016325748547
