| L(s) = 1 | + (0.528 + 1.34i)3-s + (0.0684 − 0.913i)5-s + (0.971 + 1.68i)7-s + (0.666 − 0.618i)9-s + (0.100 + 0.439i)11-s + (2.90 + 1.97i)13-s + (1.26 − 0.390i)15-s + (0.142 + 1.90i)17-s + (−2.97 − 2.76i)19-s + (−1.75 + 2.19i)21-s + (1.77 + 0.546i)23-s + (4.11 + 0.620i)25-s + (5.09 + 2.45i)27-s + (−1.90 + 4.86i)29-s + (−0.920 + 0.138i)31-s + ⋯ |
| L(s) = 1 | + (0.304 + 0.777i)3-s + (0.0306 − 0.408i)5-s + (0.367 + 0.636i)7-s + (0.222 − 0.206i)9-s + (0.0302 + 0.132i)11-s + (0.804 + 0.548i)13-s + (0.326 − 0.100i)15-s + (0.0345 + 0.461i)17-s + (−0.682 − 0.633i)19-s + (−0.382 + 0.479i)21-s + (0.369 + 0.113i)23-s + (0.822 + 0.124i)25-s + (0.980 + 0.471i)27-s + (−0.354 + 0.902i)29-s + (−0.165 + 0.0249i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63042 + 0.840841i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63042 + 0.840841i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-6.30 + 1.79i)T \) |
| good | 3 | \( 1 + (-0.528 - 1.34i)T + (-2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (-0.0684 + 0.913i)T + (-4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (-0.971 - 1.68i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.100 - 0.439i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-2.90 - 1.97i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-0.142 - 1.90i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (2.97 + 2.76i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.77 - 0.546i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (1.90 - 4.86i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (0.920 - 0.138i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (-0.277 + 0.480i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.111 - 0.139i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (1.72 - 7.57i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (8.52 - 5.81i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (9.19 + 4.42i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (5.55 + 0.837i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (-0.807 - 0.748i)T + (5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-5.41 + 1.67i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-11.7 - 7.99i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (1.13 + 1.95i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.11 + 13.0i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-0.143 - 0.365i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (3.76 + 16.4i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−10.72613151730873831743304992979, −9.469250869037762743236812958751, −9.004515841246747768613671664386, −8.350769689618756517436912384133, −7.06034310147053797139845445075, −6.07113892981929868356466310114, −4.93323129625247881395870099534, −4.20995433350884531319554474125, −3.07836271993428448405802308432, −1.54912299764834214359528077905,
1.11330683257093343561864648825, 2.41071787496236352339904479333, 3.66177102215113804337589135169, 4.80077318953985785682934769671, 6.10152076684573865094239668531, 6.90688058513791288479257867974, 7.77444600534287812761618385778, 8.309014292343099949342164804727, 9.470660490489126084162653463901, 10.64267617889660697276010378514
