L(s) = 1 | + (0.127 − 0.279i)3-s + (0.654 − 0.755i)5-s + (−2.94 + 1.89i)7-s + (1.90 + 2.19i)9-s + (−0.139 + 0.967i)11-s + (5.19 + 3.33i)13-s + (−0.127 − 0.279i)15-s + (1.70 + 0.499i)17-s + (7.80 − 2.29i)19-s + (0.153 + 1.06i)21-s + (−4.53 − 1.55i)23-s + (−0.142 − 0.989i)25-s + (1.74 − 0.511i)27-s + (2.15 + 0.631i)29-s + (−1.59 − 3.48i)31-s + ⋯ |
L(s) = 1 | + (0.0737 − 0.161i)3-s + (0.292 − 0.337i)5-s + (−1.11 + 0.715i)7-s + (0.634 + 0.731i)9-s + (−0.0419 + 0.291i)11-s + (1.43 + 0.925i)13-s + (−0.0329 − 0.0721i)15-s + (0.412 + 0.121i)17-s + (1.79 − 0.525i)19-s + (0.0334 + 0.232i)21-s + (−0.945 − 0.324i)23-s + (−0.0284 − 0.197i)25-s + (0.335 − 0.0984i)27-s + (0.399 + 0.117i)29-s + (−0.285 − 0.625i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43481 + 0.358414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43481 + 0.358414i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (4.53 + 1.55i)T \) |
good | 3 | \( 1 + (-0.127 + 0.279i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (2.94 - 1.89i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.139 - 0.967i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-5.19 - 3.33i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.70 - 0.499i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-7.80 + 2.29i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.15 - 0.631i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.59 + 3.48i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (2.03 + 2.34i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (1.16 - 1.34i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (3.43 - 7.51i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 5.00T + 47T^{2} \) |
| 53 | \( 1 + (8.36 - 5.37i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.706 - 0.454i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (0.0599 + 0.131i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.387 - 2.69i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.03 - 7.22i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-5.82 + 1.70i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (7.55 + 4.85i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (9.01 + 10.4i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-6.41 + 14.0i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (6.58 - 7.59i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.18661819290427874466446584792, −9.990817300799811773275935980582, −9.433259504648254527272029699210, −8.525656244494482116760390474061, −7.42886374129829752177853941740, −6.39947267535430412034966243744, −5.58505692973954074064231011859, −4.31492493709289727649278196244, −3.01547901175112443587170738297, −1.57887577450139976962937135272,
1.07492571396184527172714474383, 3.37281653977510473120607855681, 3.62236908203044377137318615640, 5.47175875234737932465002062070, 6.34739491026933210293285788965, 7.19129977519442898983240396434, 8.253652417174415669903725645134, 9.495404787536324212155113067346, 10.03427320782212044926995934690, 10.73296029065094527510056369385