L(s) = 1 | + (0.130 + 0.991i)2-s + (−0.991 + 0.130i)3-s + (−0.965 + 0.258i)4-s + (−0.258 − 0.965i)6-s + (−0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (1.78 + 0.882i)11-s + (0.923 − 0.382i)12-s + (0.866 − 0.5i)16-s + (−0.608 − 0.793i)17-s + (0.382 + 0.923i)18-s + (−0.739 + 1.78i)19-s + (−0.641 + 1.88i)22-s + (0.5 + 0.866i)24-s + (0.793 + 0.608i)25-s + ⋯ |
L(s) = 1 | + (0.130 + 0.991i)2-s + (−0.991 + 0.130i)3-s + (−0.965 + 0.258i)4-s + (−0.258 − 0.965i)6-s + (−0.382 − 0.923i)8-s + (0.965 − 0.258i)9-s + (1.78 + 0.882i)11-s + (0.923 − 0.382i)12-s + (0.866 − 0.5i)16-s + (−0.608 − 0.793i)17-s + (0.382 + 0.923i)18-s + (−0.739 + 1.78i)19-s + (−0.641 + 1.88i)22-s + (0.5 + 0.866i)24-s + (0.793 + 0.608i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7595728897\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7595728897\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.130 - 0.991i)T \) |
| 3 | \( 1 + (0.991 - 0.130i)T \) |
| 17 | \( 1 + (0.608 + 0.793i)T \) |
good | 5 | \( 1 + (-0.793 - 0.608i)T^{2} \) |
| 7 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 11 | \( 1 + (-1.78 - 0.882i)T + (0.608 + 0.793i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.739 - 1.78i)T + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 29 | \( 1 + (0.130 - 0.991i)T^{2} \) |
| 31 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 37 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 41 | \( 1 + (0.0983 + 0.0862i)T + (0.130 + 0.991i)T^{2} \) |
| 43 | \( 1 + (0.741 - 0.965i)T + (-0.258 - 0.965i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.258 - 0.0340i)T + (0.965 + 0.258i)T^{2} \) |
| 61 | \( 1 + (-0.793 + 0.608i)T^{2} \) |
| 67 | \( 1 + (-1.37 - 0.793i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 73 | \( 1 + (-1.75 + 0.349i)T + (0.923 - 0.382i)T^{2} \) |
| 79 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 83 | \( 1 + (0.758 - 0.0999i)T + (0.965 - 0.258i)T^{2} \) |
| 89 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 97 | \( 1 + (0.991 - 0.869i)T + (0.130 - 0.991i)T^{2} \) |
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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
−9.885883287156701882071153681679, −9.480974313774381717086626140723, −8.533418554743848193384109679632, −7.44910803127203922689799449509, −6.65930173090793205121409204309, −6.27731833170315496842467057419, −5.18934593478049139825199331900, −4.39497108048035009747711298752, −3.69618442833598835571374720436, −1.45783408385176911054375619124,
0.836786292120324365534751213634, 2.09865433863225657375618903637, 3.58788630847082861228297076850, 4.38257155511252400511555702828, 5.20953900611009756554238900089, 6.40488018392974930046875820294, 6.71378258112847289815422415768, 8.377570821129015667605562038642, 8.964143963478008190147929439452, 9.791389052749311862207033069264