| L(s) = 1 | + (0.412 − 0.0196i)2-s + (−1.82 + 0.173i)4-s + (−2.81 − 2.68i)5-s + (0.539 + 2.79i)7-s + (−1.56 + 0.224i)8-s + (−1.21 − 1.05i)10-s + (1.68 − 0.869i)11-s + (2.45 + 0.473i)13-s + (0.277 + 1.14i)14-s + (2.95 − 0.569i)16-s + (2.19 + 4.81i)17-s + (2.26 + 1.03i)19-s + (5.59 + 4.40i)20-s + (0.678 − 0.391i)22-s + (−1.47 + 4.56i)23-s + ⋯ |
| L(s) = 1 | + (0.291 − 0.0138i)2-s + (−0.910 + 0.0869i)4-s + (−1.26 − 1.20i)5-s + (0.203 + 1.05i)7-s + (−0.553 + 0.0795i)8-s + (−0.383 − 0.332i)10-s + (0.508 − 0.262i)11-s + (0.681 + 0.131i)13-s + (0.0741 + 0.305i)14-s + (0.738 − 0.142i)16-s + (0.533 + 1.16i)17-s + (0.519 + 0.237i)19-s + (1.25 + 0.984i)20-s + (0.144 − 0.0834i)22-s + (−0.307 + 0.951i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.835195 + 0.449796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.835195 + 0.449796i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (1.47 - 4.56i)T \) |
| good | 2 | \( 1 + (-0.412 + 0.0196i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (2.81 + 2.68i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-0.539 - 2.79i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (-1.68 + 0.869i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-2.45 - 0.473i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-2.19 - 4.81i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.26 - 1.03i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.293 - 3.07i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (7.06 - 5.55i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.48 + 5.06i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.86 + 4.05i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (0.990 - 1.25i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (-3.02 - 1.74i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.68 - 5.40i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.385 + 2.00i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (-5.11 + 12.7i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (4.41 - 8.55i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (-0.382 - 0.595i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.51 - 3.31i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (8.46 - 2.92i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (9.59 - 9.15i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (2.44 - 16.9i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-17.4 - 4.23i)T + (86.2 + 44.4i)T^{2} \) |
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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
−11.00890379792754307122796348174, −9.523781736422721433310916349527, −8.767681603606699886808783659131, −8.450970293346089337941483414469, −7.50552663512821178312816087745, −5.76032984388795916801302855572, −5.28397000450451278212032662915, −4.04177779396894783457277260830, −3.55708767877684102442196030283, −1.27829181802266275569567935592,
0.58462847657916639642798867875, 3.06479857442673213970505814632, 3.92979703916550056831896435567, 4.52688038530750017076480566349, 5.97613495461755816672974251132, 7.13906288347420124011797820703, 7.63057195615301861292013117420, 8.634355861790218900731661096263, 9.776321579387057189880090950542, 10.50131153247072289316095083210
