L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.230 − 0.398i)5-s − 0.999·8-s + 0.460·10-s + (−1.82 − 3.15i)11-s + (−0.730 + 1.26i)13-s + (−0.5 − 0.866i)16-s + 3.73·17-s − 4.05·19-s + (0.230 + 0.398i)20-s + (1.82 − 3.15i)22-s + (0.566 − 0.981i)23-s + (2.39 + 4.14i)25-s − 1.46·26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.102 − 0.178i)5-s − 0.353·8-s + 0.145·10-s + (−0.549 − 0.952i)11-s + (−0.202 + 0.350i)13-s + (−0.125 − 0.216i)16-s + 0.905·17-s − 0.930·19-s + (0.0514 + 0.0891i)20-s + (0.388 − 0.673i)22-s + (0.118 − 0.204i)23-s + (0.478 + 0.829i)25-s − 0.286·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.991953159\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.991953159\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.230 + 0.398i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.82 + 3.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.730 - 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3.73T + 17T^{2} \) |
| 19 | \( 1 + 4.05T + 19T^{2} \) |
| 23 | \( 1 + (-0.566 + 0.981i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.48 - 7.77i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.257 - 0.445i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.10T + 37T^{2} \) |
| 41 | \( 1 + (0.472 - 0.819i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.16 + 2.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + (-6.44 + 11.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.04 - 10.4i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.16 + 2.00i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + (-2.50 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.32 - 5.75i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.72T + 89T^{2} \) |
| 97 | \( 1 + (-5.59 - 9.68i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.746231106058422677328540429605, −8.298320235750921563274138435608, −7.43089791221656610429338452379, −6.68940754901826558957741347460, −5.86635428348070796114788435232, −5.23144896526195703117997717835, −4.40838995269031300970047171968, −3.40950710913873061793348099347, −2.57273450342307717096804763365, −1.00357531323638112860756686465,
0.71740705587828472688368475774, 2.19617762459763050916397031598, 2.73266732570780981515795935139, 3.97178420989055702517941403302, 4.60914698913755515801109817864, 5.52516218565997918063948779729, 6.24615314027148466242354607957, 7.21696786530366661834799068688, 7.953740456715247506335640201112, 8.761558926897899306942999527693