| L(s) = 1 | + (−0.976 + 1.02i)2-s + (−0.987 − 0.156i)3-s + (−0.0922 − 1.99i)4-s + (−0.698 − 2.12i)5-s + (1.12 − 0.857i)6-s + (−1.04 + 1.04i)7-s + (2.13 + 1.85i)8-s + (0.951 + 0.309i)9-s + (2.85 + 1.36i)10-s + (−4.52 + 1.47i)11-s + (−0.221 + 1.98i)12-s + (2.54 + 4.99i)13-s + (−0.0481 − 2.08i)14-s + (0.357 + 2.20i)15-s + (−3.98 + 0.368i)16-s + (0.262 + 1.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.690 + 0.723i)2-s + (−0.570 − 0.0903i)3-s + (−0.0461 − 0.998i)4-s + (−0.312 − 0.949i)5-s + (0.459 − 0.350i)6-s + (−0.394 + 0.394i)7-s + (0.754 + 0.656i)8-s + (0.317 + 0.103i)9-s + (0.902 + 0.430i)10-s + (−1.36 + 0.443i)11-s + (−0.0639 + 0.573i)12-s + (0.705 + 1.38i)13-s + (−0.0128 − 0.557i)14-s + (0.0922 + 0.569i)15-s + (−0.995 + 0.0921i)16-s + (0.0635 + 0.401i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.115798 + 0.332880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.115798 + 0.332880i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.976 - 1.02i)T \) |
| 3 | \( 1 + (0.987 + 0.156i)T \) |
| 5 | \( 1 + (0.698 + 2.12i)T \) |
| good | 7 | \( 1 + (1.04 - 1.04i)T - 7iT^{2} \) |
| 11 | \( 1 + (4.52 - 1.47i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.54 - 4.99i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.262 - 1.65i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-2.76 - 2.01i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.62 - 7.10i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (4.76 + 6.56i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.29 + 1.77i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.10 + 0.565i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (1.73 - 5.32i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (2.44 + 2.44i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.899 - 5.67i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (0.680 - 4.29i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-3.92 + 12.0i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.15 - 6.62i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.39 - 1.17i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-7.39 - 10.1i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (9.08 + 4.63i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-7.49 + 5.44i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (0.974 + 6.15i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.92 + 0.626i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (3.36 + 0.532i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.92146713647481566578584875363, −11.20895013780733921869565288553, −9.895353274672809883583350941058, −9.345179796224921353833089567556, −8.149352519744158323924422598955, −7.48464567374062595983760306027, −6.11454299200448127673481116091, −5.42365480855781006789815902104, −4.24613912624472229127241775912, −1.69436930537624372759850163744,
0.34535878442742999352209454484, 2.75098519897077608038444186367, 3.63562273058127393624266705199, 5.30322947277763806253665022029, 6.69223439829270957149038182366, 7.63095113773829148746659918673, 8.472987871752325886828250256742, 9.976154133682530239148300417518, 10.57659779811549326709865853893, 10.98848692512145335107315998300
