L(s) = 1 | + (−1 − 1.41i)3-s + 2.82i·5-s − 2.82i·7-s + (−1.00 + 2.82i)9-s − 2·11-s + 4·13-s + (4.00 − 2.82i)15-s − 5.65i·17-s + 2.82i·19-s + (−4.00 + 2.82i)21-s + 8·23-s − 3.00·25-s + (5.00 − 1.41i)27-s − 2.82i·29-s − 8.48i·31-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.816i)3-s + 1.26i·5-s − 1.06i·7-s + (−0.333 + 0.942i)9-s − 0.603·11-s + 1.10·13-s + (1.03 − 0.730i)15-s − 1.37i·17-s + 0.648i·19-s + (−0.872 + 0.617i)21-s + 1.66·23-s − 0.600·25-s + (0.962 − 0.272i)27-s − 0.525i·29-s − 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07378 - 0.555830i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07378 - 0.555830i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 5.65iT - 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 2.82iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 8.48iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 14.1iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 2.82iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 5.65iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.49581192335817737790846579189, −9.543899788964575899153031015199, −8.086919760903735334801445988177, −7.43011874599823733443574234447, −6.78366017062805758849285977129, −6.03787639292259031947254739863, −4.88512381675847195904356668940, −3.51813042541295861211098940628, −2.46031815842547419229054204497, −0.803024580290974129342567352760,
1.20454616101463948102870909768, 3.03196401929415917963456884777, 4.26639469293914258492390130553, 5.20021397482456621950003683795, 5.66347005543265168702364554483, 6.72885094213116049258362061315, 8.408413611303785018135689604177, 8.756041570822167226075603825081, 9.403001146907144055473502513772, 10.63661860865853343766449114097