| L(s) = 1 | + (−0.811 + 1.15i)2-s + (1.58 − 0.705i)3-s + (−0.683 − 1.87i)4-s + (−1.29 − 1.47i)5-s + (−0.465 + 2.40i)6-s + (1.69 + 0.222i)7-s + (2.73 + 0.732i)8-s + (2.00 − 2.23i)9-s + (2.76 − 0.302i)10-s + (2.41 + 4.89i)11-s + (−2.40 − 2.48i)12-s + (−2.03 − 6.00i)13-s + (−1.63 + 1.77i)14-s + (−3.09 − 1.42i)15-s + (−3.06 + 2.57i)16-s + (−2.66 − 2.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.573 + 0.819i)2-s + (0.913 − 0.407i)3-s + (−0.341 − 0.939i)4-s + (−0.579 − 0.661i)5-s + (−0.189 + 0.981i)6-s + (0.639 + 0.0841i)7-s + (0.965 + 0.258i)8-s + (0.667 − 0.744i)9-s + (0.874 − 0.0957i)10-s + (0.728 + 1.47i)11-s + (−0.695 − 0.718i)12-s + (−0.564 − 1.66i)13-s + (−0.435 + 0.475i)14-s + (−0.798 − 0.367i)15-s + (−0.766 + 0.642i)16-s + (−0.646 − 0.646i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35004 - 0.330748i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35004 - 0.330748i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.811 - 1.15i)T \) |
| 3 | \( 1 + (-1.58 + 0.705i)T \) |
| good | 5 | \( 1 + (1.29 + 1.47i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-1.69 - 0.222i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-2.41 - 4.89i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (2.03 + 6.00i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (2.66 + 2.66i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.0844 - 0.424i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.326 + 2.48i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-6.83 + 0.447i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-4.60 + 2.65i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0591 - 0.297i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-1.02 - 7.76i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-6.13 + 3.02i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (10.3 - 2.77i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.91 + 5.85i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-7.92 - 9.03i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.568 + 8.67i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (4.32 + 2.13i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-3.28 - 7.93i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (3.23 - 7.80i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.47 - 5.51i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (5.07 + 4.45i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (13.2 - 5.50i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-6.93 - 4.00i)T + (48.5 + 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.15074173645691339734303581519, −9.647268026396252539460384471126, −8.521645697543475038252700326807, −8.114474693639246817634498847623, −7.34034663886383685032543598339, −6.48299198715736191231319657915, −4.92116674691702561638222074419, −4.35078230066054580407080238645, −2.43844367185207619455722610573, −0.958558311383797403662201308886,
1.66509229025205226434341403125, 2.93522139213281873027860435052, 3.86417678875109672794445285208, 4.60353091982880671643521621031, 6.63597819626737548102496315306, 7.52038819737598632313814618787, 8.527035787337620455354549069187, 8.898250452155349057786372211816, 9.905538085632453238463601603541, 10.90970415818199834385925829895
