L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (1 − 2i)5-s − 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (0.707 + 2.12i)10-s − 1.96·11-s + (0.707 + 0.707i)12-s + (−2.37 + 2.37i)13-s + (0.707 + 2.12i)15-s − 1.00·16-s + (3.38 + 3.38i)17-s + (0.707 + 0.707i)18-s − 3.41·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.447 − 0.894i)5-s − 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.223 + 0.670i)10-s − 0.592·11-s + (0.204 + 0.204i)12-s + (−0.659 + 0.659i)13-s + (0.182 + 0.547i)15-s − 0.250·16-s + (0.821 + 0.821i)17-s + (0.166 + 0.166i)18-s − 0.783·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7829143313\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7829143313\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 + (2.37 - 2.37i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.38 - 3.38i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 + (3 + 3i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.22iT - 29T^{2} \) |
| 31 | \( 1 - 1.98iT - 31T^{2} \) |
| 37 | \( 1 + (-5.35 + 5.35i)T - 37iT^{2} \) |
| 41 | \( 1 - 11.5iT - 41T^{2} \) |
| 43 | \( 1 + (-7.81 - 7.81i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.93 - 4.93i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.39 - 8.39i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 - 3.92iT - 61T^{2} \) |
| 67 | \( 1 + (-2.36 + 2.36i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.14T + 71T^{2} \) |
| 73 | \( 1 + (-2.01 + 2.01i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.5iT - 79T^{2} \) |
| 83 | \( 1 + (10.9 - 10.9i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.02T + 89T^{2} \) |
| 97 | \( 1 + (-5.08 - 5.08i)T + 97iT^{2} \) |
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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
−9.639727594672841156915862267972, −9.036757876612941604264420818099, −8.205494403151110263160542026571, −7.49206023241535713159686679611, −6.26963658740201419522906232269, −5.80612722809718605037710976265, −4.80409881380762323069046942937, −4.20135381231065571145168943142, −2.48185236196017427737921236479, −1.17667895589225222122478501363,
0.42344643220803674872216500629, 2.09427694380215730711449944056, 2.74862303573892604528683585210, 3.91704895813570101330885335653, 5.31541873509206917962655732532, 5.93290841794273400539724063996, 7.10863674835007771619040429757, 7.52297165488946719099865839435, 8.389577092380421429978189185423, 9.552586529634863323177085231259