L(s) = 1 | + (1 − 1.73i)3-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s − 4·13-s + (1 − 1.73i)17-s + (3 + 5.19i)19-s + (−4 − 6.92i)23-s + (2.5 − 4.33i)25-s + 4.00·27-s + 2·29-s + (2 − 3.46i)31-s + (−3.99 − 6.92i)33-s + (−5 − 8.66i)37-s + (−4 + 6.92i)39-s − 10·41-s + ⋯ |
L(s) = 1 | + (0.577 − 0.999i)3-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s − 1.10·13-s + (0.242 − 0.420i)17-s + (0.688 + 1.19i)19-s + (−0.834 − 1.44i)23-s + (0.5 − 0.866i)25-s + 0.769·27-s + 0.371·29-s + (0.359 − 0.622i)31-s + (−0.696 − 1.20i)33-s + (−0.821 − 1.42i)37-s + (−0.640 + 1.10i)39-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.902213837\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.902213837\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.935618744639606178158548065259, −8.288160011650115988026063046662, −7.66673133820314280638966525338, −6.84796716705882411980144600197, −6.13014693493831836706981825908, −5.10182222428006508201329252193, −3.95419917458395826946989809104, −2.86626175830223330806690385101, −2.01399891433192791332222017452, −0.69972999514909345743633137145,
1.59508245037601386228854357373, 2.93991732126155711150269897491, 3.69065127170005976181181153096, 4.74148336838648099636782402493, 5.16370727224857433240921688883, 6.65495064580876221518634699169, 7.24502425935227376474024920526, 8.226742755290609853996172103233, 9.131990479126547086206410105317, 9.764664668741302425296577013520