| L(s) = 1 | + (−0.542 − 1.30i)2-s + (0.290 + 0.956i)3-s + (−1.41 + 1.41i)4-s + (0.155 + 1.57i)5-s + (1.09 − 0.898i)6-s + (0.0321 − 0.0481i)7-s + (2.61 + 1.07i)8-s + (−0.831 + 0.555i)9-s + (1.97 − 1.05i)10-s + (−4.37 + 2.33i)11-s + (−1.76 − 0.939i)12-s + (−4.81 − 0.474i)13-s + (−0.0802 − 0.0158i)14-s + (−1.46 + 0.606i)15-s + (−0.0142 − 3.99i)16-s + (0.601 + 0.249i)17-s + ⋯ |
| L(s) = 1 | + (−0.383 − 0.923i)2-s + (0.167 + 0.552i)3-s + (−0.705 + 0.708i)4-s + (0.0695 + 0.705i)5-s + (0.445 − 0.366i)6-s + (0.0121 − 0.0181i)7-s + (0.924 + 0.380i)8-s + (−0.277 + 0.185i)9-s + (0.625 − 0.334i)10-s + (−1.31 + 0.704i)11-s + (−0.509 − 0.271i)12-s + (−1.33 − 0.131i)13-s + (−0.0214 − 0.00424i)14-s + (−0.378 + 0.156i)15-s + (−0.00356 − 0.999i)16-s + (0.145 + 0.0603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0553 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0553 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.461220 + 0.487472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.461220 + 0.487472i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.542 + 1.30i)T \) |
| 3 | \( 1 + (-0.290 - 0.956i)T \) |
| good | 5 | \( 1 + (-0.155 - 1.57i)T + (-4.90 + 0.975i)T^{2} \) |
| 7 | \( 1 + (-0.0321 + 0.0481i)T + (-2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (4.37 - 2.33i)T + (6.11 - 9.14i)T^{2} \) |
| 13 | \( 1 + (4.81 + 0.474i)T + (12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-0.601 - 0.249i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (1.22 - 1.00i)T + (3.70 - 18.6i)T^{2} \) |
| 23 | \( 1 + (1.04 - 5.26i)T + (-21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (3.92 - 7.34i)T + (-16.1 - 24.1i)T^{2} \) |
| 31 | \( 1 + (0.512 - 0.512i)T - 31iT^{2} \) |
| 37 | \( 1 + (-6.25 + 7.61i)T + (-7.21 - 36.2i)T^{2} \) |
| 41 | \( 1 + (-10.1 - 2.02i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.0283 + 0.0935i)T + (-35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (2.08 - 5.03i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (3.80 + 7.11i)T + (-29.4 + 44.0i)T^{2} \) |
| 59 | \( 1 + (6.36 - 0.626i)T + (57.8 - 11.5i)T^{2} \) |
| 61 | \( 1 + (-4.53 + 1.37i)T + (50.7 - 33.8i)T^{2} \) |
| 67 | \( 1 + (3.87 - 1.17i)T + (55.7 - 37.2i)T^{2} \) |
| 71 | \( 1 + (-1.73 - 1.16i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-5.28 - 7.90i)T + (-27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (3.36 + 8.12i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-8.89 - 10.8i)T + (-16.1 + 81.4i)T^{2} \) |
| 89 | \( 1 + (2.14 + 10.7i)T + (-82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-10.4 + 10.4i)T - 97iT^{2} \) |
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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
−11.21136056244763884198828819753, −10.64202240150808702892056261717, −9.863185343508558837638335173408, −9.242270605850700132482770319375, −7.83608418881012047021080284758, −7.33193633411915999760266710554, −5.44572652543442243934248069729, −4.47658470302632688462300543150, −3.13415177244106378513124091658, −2.25666738797831814855546985911,
0.47489046781591979121359887064, 2.45985057958964832653558368367, 4.52468283418759377952334087202, 5.40218841752217768376647562828, 6.39179074978170261828386534812, 7.58809124966315153882083841904, 8.106224062000693491811491428709, 9.046819452387181934498712854722, 9.930422053706365423025287685695, 10.92657321410171289546930132726
