L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 5·7-s + 0.999·8-s + (0.499 − 0.866i)10-s + 2·11-s + (0.5 − 0.866i)13-s + (2.5 + 4.33i)14-s + (−0.5 − 0.866i)16-s + (−3 − 5.19i)17-s + (4 − 1.73i)19-s − 0.999·20-s + (−1 − 1.73i)22-s + (−2 + 3.46i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 1.88·7-s + 0.353·8-s + (0.158 − 0.273i)10-s + 0.603·11-s + (0.138 − 0.240i)13-s + (0.668 + 1.15i)14-s + (−0.125 − 0.216i)16-s + (−0.727 − 1.26i)17-s + (0.917 − 0.397i)19-s − 0.223·20-s + (−0.213 − 0.369i)22-s + (−0.417 + 0.722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9767794260\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9767794260\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 7 | \( 1 + 5T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.5 - 7.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4 - 6.92i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.431148954737263384006189958416, −9.047343765424109813048468057143, −7.72464224173532809558533035133, −6.90303539371949252280362715727, −6.36154490916606090465728920295, −5.30585709863886496390495874195, −4.02512249562686751120459775290, −3.15344074435884395831994111622, −2.58307662823097614791089256673, −0.897804706960103125377214382675,
0.56101995698894936967198449642, 2.13454315171080859297262684383, 3.52585264818415165372871366777, 4.24099311413043242897178801686, 5.55055241836944428668383661448, 6.42018481574896499679662469980, 6.54581524989275941556083397849, 7.75122450630389272081915655447, 8.637874998591582187993354139045, 9.277442539678912134180799790743