| 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
−10.64267617889660697276010378514, −9.470660490489126084162653463901, −8.309014292343099949342164804727, −7.77444600534287812761618385778, −6.90688058513791288479257867974, −6.10152076684573865094239668531, −4.80077318953985785682934769671, −3.66177102215113804337589135169, −2.41071787496236352339904479333, −1.11330683257093343561864648825,
1.54912299764834214359528077905, 3.07836271993428448405802308432, 4.20995433350884531319554474125, 4.93323129625247881395870099534, 6.07113892981929868356466310114, 7.06034310147053797139845445075, 8.350769689618756517436912384133, 9.004515841246747768613671664386, 9.469250869037762743236812958751, 10.72613151730873831743304992979
