L(s) = 1 | + (−1.06 + 1.06i)2-s + (−1.73 + 0.0150i)3-s − 0.285i·4-s + (2.03 + 0.933i)5-s + (1.83 − 1.86i)6-s + (−1.83 − 1.83i)8-s + (2.99 − 0.0520i)9-s + (−3.16 + 1.17i)10-s − 0.914i·11-s + (0.00428 + 0.493i)12-s + (−3.07 + 3.07i)13-s + (−3.53 − 1.58i)15-s + 4.48·16-s + (0.850 − 0.850i)17-s + (−3.15 + 3.26i)18-s + 6.87i·19-s + ⋯ |
L(s) = 1 | + (−0.755 + 0.755i)2-s + (−0.999 + 0.00867i)3-s − 0.142i·4-s + (0.908 + 0.417i)5-s + (0.749 − 0.762i)6-s + (−0.648 − 0.648i)8-s + (0.999 − 0.0173i)9-s + (−1.00 + 0.371i)10-s − 0.275i·11-s + (0.00123 + 0.142i)12-s + (−0.854 + 0.854i)13-s + (−0.912 − 0.409i)15-s + 1.12·16-s + (0.206 − 0.206i)17-s + (−0.742 + 0.768i)18-s + 1.57i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0336015 - 0.509330i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0336015 - 0.509330i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.73 - 0.0150i)T \) |
| 5 | \( 1 + (-2.03 - 0.933i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.06 - 1.06i)T - 2iT^{2} \) |
| 11 | \( 1 + 0.914iT - 11T^{2} \) |
| 13 | \( 1 + (3.07 - 3.07i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.850 + 0.850i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.87iT - 19T^{2} \) |
| 23 | \( 1 + (-1.38 - 1.38i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 + (-0.567 - 0.567i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.922iT - 41T^{2} \) |
| 43 | \( 1 + (4.80 - 4.80i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.41 - 7.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.79 + 7.79i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.88T + 59T^{2} \) |
| 61 | \( 1 - 1.06T + 61T^{2} \) |
| 67 | \( 1 + (5.00 + 5.00i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.557iT - 71T^{2} \) |
| 73 | \( 1 + (-1.54 + 1.54i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.03iT - 79T^{2} \) |
| 83 | \( 1 + (-2.38 - 2.38i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \) |
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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
−10.60026643681832538543967122019, −9.732496534676097960301752720981, −9.392978959156286552635888987051, −8.068705173321344384659429297097, −7.23693365001092910274650461949, −6.46564887448212362141017156589, −5.88144909890453597006326434674, −4.81099907076398731897242058464, −3.35198727178490491174436682224, −1.61477013004189210453601214155,
0.38917036580864607418244714434, 1.64704817587697636502585166521, 2.81192501876491857008040498232, 4.77673848449035409512428967574, 5.34223822328825501474615579200, 6.28445111669447476407928226438, 7.28994327332873729754051506334, 8.559140885519571525850442341146, 9.439347595949339863036751051553, 10.02220195776663875117007144760