| L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.0894 + 0.621i)3-s + (0.415 + 0.909i)4-s + (−2.40 − 0.705i)5-s + (0.260 − 0.571i)6-s + (−0.654 + 0.755i)7-s + (0.142 − 0.989i)8-s + (2.49 − 0.734i)9-s + (1.63 + 1.89i)10-s + (−2.09 + 1.34i)11-s + (−0.528 + 0.339i)12-s + (3.96 + 4.57i)13-s + (0.959 − 0.281i)14-s + (0.223 − 1.55i)15-s + (−0.654 + 0.755i)16-s + (−2.78 + 6.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (0.0516 + 0.359i)3-s + (0.207 + 0.454i)4-s + (−1.07 − 0.315i)5-s + (0.106 − 0.233i)6-s + (−0.247 + 0.285i)7-s + (0.0503 − 0.349i)8-s + (0.833 − 0.244i)9-s + (0.518 + 0.598i)10-s + (−0.632 + 0.406i)11-s + (−0.152 + 0.0980i)12-s + (1.09 + 1.26i)13-s + (0.256 − 0.0752i)14-s + (0.0577 − 0.401i)15-s + (−0.163 + 0.188i)16-s + (−0.675 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.534131 + 0.426619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.534131 + 0.426619i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.73 - 0.787i)T \) |
| good | 3 | \( 1 + (-0.0894 - 0.621i)T + (-2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (2.40 + 0.705i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (2.09 - 1.34i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.96 - 4.57i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.78 - 6.09i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.66 - 3.63i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (2.09 - 4.59i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.678 + 4.72i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (3.36 - 0.986i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-11.2 - 3.29i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.533 + 3.70i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 4.86T + 47T^{2} \) |
| 53 | \( 1 + (-5.29 + 6.11i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (5.56 + 6.41i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.02 - 7.13i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-10.1 - 6.52i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (10.9 + 7.06i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.25 - 2.75i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (6.14 + 7.08i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (4.01 - 1.17i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.62 + 11.2i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (6.98 + 2.05i)T + (81.6 + 52.4i)T^{2} \) |
| 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
−11.75044414998618378000098061695, −10.84053276396532267987810494190, −9.989788652887685354554981815393, −9.003260433242256556472355629186, −8.226316763630459488317585311548, −7.28202937356970930726184746523, −6.07736506686397561964456223124, −4.22875561693179913686170664797, −3.78587071874868770397799884548, −1.76061874218304849549892542415,
0.61020235886100333387996028087, 2.83184648823220089887747881762, 4.24149460245986148084996816397, 5.67607868881576880189185524323, 6.96666432384413764185081493556, 7.60132372918266429541966825959, 8.299588238256506748249155574185, 9.500240774042290764006297490318, 10.63530082333616537942939745697, 11.14238189719498360855482934680
