L(s) = 1 | − 3.46i·5-s + (−2 + 1.73i)7-s + 3.46i·11-s − 3.46i·13-s − 2·19-s + 3.46i·23-s − 6.99·25-s + 6·29-s − 8·31-s + (5.99 + 6.92i)35-s + 2·37-s + 6.92i·41-s + 10.3i·43-s + (1.00 − 6.92i)49-s + 6·53-s + ⋯ |
L(s) = 1 | − 1.54i·5-s + (−0.755 + 0.654i)7-s + 1.04i·11-s − 0.960i·13-s − 0.458·19-s + 0.722i·23-s − 1.39·25-s + 1.11·29-s − 1.43·31-s + (1.01 + 1.17i)35-s + 0.328·37-s + 1.08i·41-s + 1.58i·43-s + (0.142 − 0.989i)49-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.203222481\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.203222481\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.475164310589725539535958914352, −8.019826144864740407884777297903, −7.09734154702325248906431418607, −6.17887122719955139873575357224, −5.44434021201073622800100497731, −4.87582852841394568726159052194, −4.07564610991572297783634717864, −3.03635093879846425250562422419, −2.00298502874942725079083606883, −0.926629528666825442809814410938,
0.41472991974883482330920704373, 2.07195135518234775470214142646, 2.94825323482482435308327660990, 3.64549085750344638659901625137, 4.29330293750650602807929625854, 5.67239664290434402623655635383, 6.27874542988212070303393365281, 7.04821121628172993232073123278, 7.19223352513606076732536581838, 8.438706800639572008712255992784