L(s) = 1 | + (−0.866 − 1.5i)5-s + (2.62 + 0.358i)7-s + (3.67 + 2.12i)11-s + (2.12 − 1.22i)13-s + 1.73·17-s − 2.44i·19-s + (−7.34 + 4.24i)23-s + (1 − 1.73i)25-s + (3.67 + 2.12i)29-s + (6.36 − 3.67i)31-s + (−1.73 − 4.24i)35-s + 37-s + (0.866 + 1.5i)41-s + (−3.5 + 6.06i)43-s + (−6.06 + 10.5i)47-s + ⋯ |
L(s) = 1 | + (−0.387 − 0.670i)5-s + (0.990 + 0.135i)7-s + (1.10 + 0.639i)11-s + (0.588 − 0.339i)13-s + 0.420·17-s − 0.561i·19-s + (−1.53 + 0.884i)23-s + (0.200 − 0.346i)25-s + (0.682 + 0.393i)29-s + (1.14 − 0.659i)31-s + (−0.292 − 0.717i)35-s + 0.164·37-s + (0.135 + 0.234i)41-s + (−0.533 + 0.924i)43-s + (−0.884 + 1.53i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.100787814\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.100787814\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 - 0.358i)T \) |
good | 5 | \( 1 + (0.866 + 1.5i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.67 - 2.12i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + (7.34 - 4.24i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.67 - 2.12i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.36 + 3.67i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + (-0.866 - 1.5i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.06 - 10.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (4.33 + 7.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.12 - 1.22i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.06 + 10.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + (2.12 + 1.22i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.922742185507355540765900925872, −8.077334251512244035145003064784, −7.78539710769445053655575100414, −6.57264479951270218368978449932, −5.87744450951853073329620572941, −4.70243605006127652367890121058, −4.40974526245621192312464677525, −3.28258263064797625831044655545, −1.87994450247570430619104476806, −0.989537306630174623499633875489,
1.04332160180897363888189693860, 2.16216586667196662418533876146, 3.49127029445457978203840416712, 4.02862830689135906803633496659, 5.02927121510368850743117948226, 6.13407955626275235769231175114, 6.63554425434318225651849892287, 7.59001107090740510887376750857, 8.381375088381328349878521283520, 8.754673682809188036905960127720