L(s) = 1 | + (1.71 − 0.206i)3-s − 4.18·5-s + (2.91 − 0.711i)9-s + 1.42i·11-s + (0.850 − 0.491i)13-s + (−7.19 + 0.866i)15-s + (0.185 + 0.321i)17-s + (−4.30 − 2.48i)19-s + 5.75i·23-s + 12.5·25-s + (4.86 − 1.82i)27-s + (7.31 + 4.22i)29-s + (6.28 + 3.62i)31-s + (0.294 + 2.44i)33-s + (−1.73 + 3.00i)37-s + ⋯ |
L(s) = 1 | + (0.992 − 0.119i)3-s − 1.87·5-s + (0.971 − 0.237i)9-s + 0.429i·11-s + (0.235 − 0.136i)13-s + (−1.85 + 0.223i)15-s + (0.0449 + 0.0779i)17-s + (−0.988 − 0.570i)19-s + 1.20i·23-s + 2.50·25-s + (0.936 − 0.351i)27-s + (1.35 + 0.784i)29-s + (1.12 + 0.651i)31-s + (0.0512 + 0.425i)33-s + (−0.284 + 0.493i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.647689601\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.647689601\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 + 0.206i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 4.18T + 5T^{2} \) |
| 11 | \( 1 - 1.42iT - 11T^{2} \) |
| 13 | \( 1 + (-0.850 + 0.491i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.185 - 0.321i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.30 + 2.48i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.75iT - 23T^{2} \) |
| 29 | \( 1 + (-7.31 - 4.22i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.28 - 3.62i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.73 - 3.00i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.06 - 1.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.00 + 5.21i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.13 - 7.16i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.30 - 2.48i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.27 + 3.94i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.50 + 3.75i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.03 + 8.71i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (8.25 - 4.76i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.25 - 7.37i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.972 - 1.68i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.90 - 6.76i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.34 - 1.92i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.098462600921621190151863269822, −8.402580394259178176919751986135, −7.979888184197476059444615663905, −7.14446803324142779729810627593, −6.61026073550280351412482024038, −4.91623347674613350939513819640, −4.23792674439153780268572252131, −3.47947607103633684848048257854, −2.66670454196793040020472467650, −1.08899242454003978017365315579,
0.67773504726193214925363301324, 2.43952757869804375458311033338, 3.34863833582913558749971172678, 4.21203492567175447773539459879, 4.56769765409827209221714649756, 6.22116384862032284824790589443, 7.07476006431389234222869630550, 7.87082353058700444268432997335, 8.446467150258650958321200319156, 8.763558899807737473711840232939