L(s) = 1 | + (0.618 − 1.07i)3-s + (1.61 + 2.80i)5-s + (0.736 + 1.27i)9-s + (3.23 − 5.60i)11-s − 0.763·13-s + 4·15-s + (2.23 − 3.87i)17-s + (−0.618 − 1.07i)19-s + (−2 − 3.46i)23-s + (−2.73 + 4.73i)25-s + 5.52·27-s − 4.47·29-s + (−1.23 + 2.14i)31-s + (−3.99 − 6.92i)33-s + (2.23 + 3.87i)37-s + ⋯ |
L(s) = 1 | + (0.356 − 0.618i)3-s + (0.723 + 1.25i)5-s + (0.245 + 0.424i)9-s + (0.975 − 1.68i)11-s − 0.211·13-s + 1.03·15-s + (0.542 − 0.939i)17-s + (−0.141 − 0.245i)19-s + (−0.417 − 0.722i)23-s + (−0.547 + 0.947i)25-s + 1.06·27-s − 0.830·29-s + (−0.222 + 0.384i)31-s + (−0.696 − 1.20i)33-s + (0.367 + 0.636i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.430530251\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.430530251\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.618 + 1.07i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.61 - 2.80i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.23 + 5.60i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 + (-2.23 + 3.87i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.618 + 1.07i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + (1.23 - 2.14i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + (-5.23 - 9.06i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.61 - 7.99i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.61 - 9.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + (1.47 - 2.54i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.47 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−9.337668940882816269874739626295, −8.624726065687283472881688273242, −7.61859524478455111339523926514, −7.03212659754555051377460567314, −6.21096437715703021896409158587, −5.63322644018677180133594010184, −4.20086765688749714920559095768, −3.01130244317053824634602620685, −2.47452599232644446512216152154, −1.12119898595768259774953155189,
1.27834936501295305893782705802, 2.14106352375399772421463332286, 3.91358325363214017269895142489, 4.21340540097511975254601781891, 5.27750310080199303799399355696, 6.04699202568584551542536949452, 7.11196009201750804713482240043, 7.981649269363956728136887973432, 9.072059975504621395228337445566, 9.428223225127409840227544167436