| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.654 + 0.755i)5-s + (−0.959 − 0.281i)6-s + (−2.43 + 1.56i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (0.841 + 0.540i)10-s + (−0.314 + 2.18i)11-s + (−0.142 + 0.989i)12-s + (3.10 + 1.99i)13-s + (1.89 + 2.19i)14-s + (0.415 + 0.909i)15-s + (0.841 − 0.540i)16-s + (5.05 + 1.48i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (−0.292 + 0.337i)5-s + (−0.391 − 0.115i)6-s + (−0.921 + 0.592i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (0.266 + 0.170i)10-s + (−0.0948 + 0.659i)11-s + (−0.0410 + 0.285i)12-s + (0.859 + 0.552i)13-s + (0.507 + 0.585i)14-s + (0.107 + 0.234i)15-s + (0.210 − 0.135i)16-s + (1.22 + 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25592 - 0.106293i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25592 - 0.106293i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (0.0513 + 4.79i)T \) |
| good | 7 | \( 1 + (2.43 - 1.56i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.314 - 2.18i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-3.10 - 1.99i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-5.05 - 1.48i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-6.48 + 1.90i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.668 - 0.196i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.94 - 6.44i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.74 - 3.17i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (5.15 - 5.95i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (4.08 - 8.93i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 6.60T + 47T^{2} \) |
| 53 | \( 1 + (-7.40 + 4.75i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.98 - 2.56i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.67 - 12.4i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (1.92 + 13.3i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (0.532 + 3.70i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (8.25 - 2.42i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-10.3 - 6.63i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (10.2 + 11.7i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.02 + 6.62i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-4.62 + 5.33i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.28329363863387210942616970329, −9.727340599415438099537276217566, −8.790051631606348031885914077739, −7.982456771302613910787545450857, −6.92253043889509628375441583503, −6.12531246903806507940648394238, −4.84313161909311407519884716303, −3.43021822016467137212193977547, −2.82230775921799402573016504105, −1.31306765970567643073738011979,
0.796231054374155336841029883142, 3.34596550904249486066443418556, 3.75584231121614097703680592453, 5.30307275292081252411831136732, 5.84341194719815097222508417149, 7.13565238844331784876739854529, 7.891965036529101272393913078967, 8.673144394987388884869734077168, 9.749373013549616791606196290657, 10.05626427963829269625882532456
