| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)6-s + (0.487 − 0.844i)7-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.939 + 0.342i)10-s + (−1.31 − 2.27i)11-s + (−0.5 + 0.866i)12-s + (1.09 − 6.20i)13-s + (−0.747 + 0.627i)14-s + (0.766 + 0.642i)15-s + (0.173 + 0.984i)16-s + (−2.43 − 0.886i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.100 + 0.568i)3-s + (0.383 + 0.321i)4-s + (0.342 − 0.287i)5-s + (0.0708 − 0.402i)6-s + (0.184 − 0.319i)7-s + (−0.176 − 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.297 + 0.108i)10-s + (−0.396 − 0.686i)11-s + (−0.144 + 0.249i)12-s + (0.303 − 1.72i)13-s + (−0.199 + 0.167i)14-s + (0.197 + 0.165i)15-s + (0.0434 + 0.246i)16-s + (−0.590 − 0.214i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.950737 - 0.523995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.950737 - 0.523995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-4.21 + 1.09i)T \) |
| good | 7 | \( 1 + (-0.487 + 0.844i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.31 + 2.27i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.09 + 6.20i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.43 + 0.886i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.535 - 0.449i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (0.522 - 0.190i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.09 - 3.63i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.66T + 37T^{2} \) |
| 41 | \( 1 + (0.676 + 3.83i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.60 + 5.54i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.19 + 1.88i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (9.11 + 7.64i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.56 - 1.29i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.76 - 4.83i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.53 + 0.921i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (2.57 - 2.15i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.04 + 11.5i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.24 - 7.04i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.16 + 7.21i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.26 - 7.19i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (14.0 + 5.12i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.65707743136978085361777042165, −9.744435009010859403286749466980, −8.910887608259595411108790851779, −8.143135507167887197806275131311, −7.29361150782637357079844471273, −5.86712158314024406394413825163, −5.10194858597455223275712784570, −3.64335527254071529168651592071, −2.64586394498384204568794446140, −0.796153671970059959307306676533,
1.59717600169918437894375539280, 2.56545637216652150530285413414, 4.31618282658944448944747695631, 5.65817356449639115907539773987, 6.57482151086962481499459526711, 7.29311076884032299722886036333, 8.181842239019232362745697997267, 9.232983311108819870224896732169, 9.679757785897463092976939627605, 10.95965512960928412675234949397
