L(s) = 1 | + (−1.42 − 0.985i)3-s + (−0.766 + 0.766i)5-s + 7-s + (1.05 + 2.80i)9-s + (−3.09 − 3.09i)11-s + (0.262 − 0.262i)13-s + (1.84 − 0.336i)15-s + 2.68i·17-s + (2.44 + 2.44i)19-s + (−1.42 − 0.985i)21-s − 3.24i·23-s + 3.82i·25-s + (1.25 − 5.04i)27-s + (−3.28 − 3.28i)29-s + 3.76i·31-s + ⋯ |
L(s) = 1 | + (−0.822 − 0.568i)3-s + (−0.342 + 0.342i)5-s + 0.377·7-s + (0.353 + 0.935i)9-s + (−0.932 − 0.932i)11-s + (0.0728 − 0.0728i)13-s + (0.476 − 0.0869i)15-s + 0.651i·17-s + (0.559 + 0.559i)19-s + (−0.310 − 0.214i)21-s − 0.676i·23-s + 0.764i·25-s + (0.241 − 0.970i)27-s + (−0.610 − 0.610i)29-s + 0.676i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.050799857\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.050799857\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.42 + 0.985i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (0.766 - 0.766i)T - 5iT^{2} \) |
| 11 | \( 1 + (3.09 + 3.09i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.262 + 0.262i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.68iT - 17T^{2} \) |
| 19 | \( 1 + (-2.44 - 2.44i)T + 19iT^{2} \) |
| 23 | \( 1 + 3.24iT - 23T^{2} \) |
| 29 | \( 1 + (3.28 + 3.28i)T + 29iT^{2} \) |
| 31 | \( 1 - 3.76iT - 31T^{2} \) |
| 37 | \( 1 + (-4.88 - 4.88i)T + 37iT^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + (-0.938 + 0.938i)T - 43iT^{2} \) |
| 47 | \( 1 + 0.764T + 47T^{2} \) |
| 53 | \( 1 + (4.14 - 4.14i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.63 - 2.63i)T + 59iT^{2} \) |
| 61 | \( 1 + (-10.7 + 10.7i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8.41 - 8.41i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.7iT - 71T^{2} \) |
| 73 | \( 1 + 6.79iT - 73T^{2} \) |
| 79 | \( 1 + 3.83iT - 79T^{2} \) |
| 83 | \( 1 + (-0.107 + 0.107i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.00T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−9.846954518471279448792688388831, −8.542156603474616848863469667374, −7.88303091174772770164400833135, −7.28904726620825781134079171663, −6.18323593668292351878820840355, −5.63588191845893100398946191055, −4.70224766687412322546681427207, −3.51305269871411871567951205055, −2.30965701990797436183952834099, −0.901849745892725457742734464840,
0.67268764695250543614383388301, 2.34931071881552510744904236763, 3.75091952672553536422638216232, 4.69873348961432063645321036050, 5.18292580043916853014569213849, 6.13001781620674768623671890540, 7.29447597564811576067250280672, 7.76644347732730349690522687328, 9.030166555511237374875616148729, 9.623989805612063038674745551492