L(s) = 1 | + (1.61 + 0.618i)3-s + 5-s + (0.381 + 2.61i)7-s + (2.23 + 2.00i)9-s − 0.763i·11-s − 1.23i·13-s + (1.61 + 0.618i)15-s − 4.47·17-s + 7.23i·19-s + (−1.00 + 4.47i)21-s + 7.70i·23-s + 25-s + (2.38 + 4.61i)27-s − 4i·29-s − 3.23i·31-s + ⋯ |
L(s) = 1 | + (0.934 + 0.356i)3-s + 0.447·5-s + (0.144 + 0.989i)7-s + (0.745 + 0.666i)9-s − 0.230i·11-s − 0.342i·13-s + (0.417 + 0.159i)15-s − 1.08·17-s + 1.66i·19-s + (−0.218 + 0.975i)21-s + 1.60i·23-s + 0.200·25-s + (0.458 + 0.888i)27-s − 0.742i·29-s − 0.581i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.545374703\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.545374703\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.61 - 0.618i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-0.381 - 2.61i)T \) |
good | 11 | \( 1 + 0.763iT - 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 7.23iT - 19T^{2} \) |
| 23 | \( 1 - 7.70iT - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 3.23iT - 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 + 9.23iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 4.94iT - 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 6.76iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 14.1iT - 97T^{2} \) |
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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
−9.540969387492580475372130872440, −8.745154809358580873267726965530, −8.104119879820808382645262252799, −7.38329895008111312607599302052, −6.08748149738753223274716933103, −5.55387481853556264941821087030, −4.43619079394787632133460558110, −3.51017889261060928663779038551, −2.50096369320823543103859402749, −1.70612688346468317981159037595,
0.871798362609212949307305579397, 2.15999839400117362221158975062, 2.95292182682839107690267825389, 4.27821513789234498385610297349, 4.68961519967663045300257212291, 6.27870688880799718606298737805, 6.94629712971177390308030646983, 7.43972609930148731431236386392, 8.609758191903891171204826646257, 8.984802923773237836354264678631