L(s) = 1 | + 1.73i·7-s + (4.5 − 2.59i)13-s + (−4 − 1.73i)19-s + (2.5 + 4.33i)25-s + 7·31-s − 5.19i·37-s + (1.5 + 0.866i)43-s + 4·49-s + (−0.5 − 0.866i)61-s + (5.5 + 9.52i)67-s + (8.5 − 14.7i)73-s + (−8.5 + 14.7i)79-s + (4.5 + 7.79i)91-s + (12 + 6.92i)97-s − 7·103-s + ⋯ |
L(s) = 1 | + 0.654i·7-s + (1.24 − 0.720i)13-s + (−0.917 − 0.397i)19-s + (0.5 + 0.866i)25-s + 1.25·31-s − 0.854i·37-s + (0.228 + 0.132i)43-s + 0.571·49-s + (−0.0640 − 0.110i)61-s + (0.671 + 1.16i)67-s + (0.994 − 1.72i)73-s + (−0.956 + 1.65i)79-s + (0.471 + 0.817i)91-s + (1.21 + 0.703i)97-s − 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.900212836\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.900212836\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 5.19iT - 37T^{2} \) |
| 41 | \( 1 + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 - 0.866i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.5 + 14.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12 - 6.92i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.719140888055867441608011788356, −8.306474864771470966562884629388, −7.35411392181660839900008508374, −6.43428391563242308233867846327, −5.83454040499677930876511298497, −5.03376896420342561840532107718, −4.03817414099960516379896148426, −3.13815437477491998902486821030, −2.20334430234549329988242126216, −0.910470052617604753593672075841,
0.855825255866622988987925164973, 1.98287532879091870427986315871, 3.19214276063401651569658488838, 4.12763693954251826797621201517, 4.65295640390820777370049660403, 5.93118469062646253544049786026, 6.48565859301277859419015297691, 7.18380769215877349521827336958, 8.295281912548680510010605338664, 8.551437021430137707522251499780