L(s) = 1 | + (1 + 1.73i)5-s + (−2 + 3.46i)7-s + (−2 + 3.46i)11-s + (1 + 1.73i)13-s + 6·17-s + 4·19-s + (0.500 − 0.866i)25-s + (1 − 1.73i)29-s + (2 + 3.46i)31-s − 7.99·35-s − 2·37-s + (1 + 1.73i)41-s + (2 − 3.46i)43-s + (−4 + 6.92i)47-s + (−4.49 − 7.79i)49-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (−0.755 + 1.30i)7-s + (−0.603 + 1.04i)11-s + (0.277 + 0.480i)13-s + 1.45·17-s + 0.917·19-s + (0.100 − 0.173i)25-s + (0.185 − 0.321i)29-s + (0.359 + 0.622i)31-s − 1.35·35-s − 0.328·37-s + (0.156 + 0.270i)41-s + (0.304 − 0.528i)43-s + (−0.583 + 1.01i)47-s + (−0.642 − 1.11i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.561893445\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.561893445\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.353773918829635163119583115159, −8.466770107578634234459457748071, −7.55126874611262419808969959611, −6.86166440945316610047332621942, −6.01006486638016017582975513082, −5.51606420923729636508185604890, −4.51376439229060954432598363816, −3.11594811809659479814376814325, −2.75777918642639635223666845750, −1.61243274288161144511544500488,
0.54623318411897654023694048051, 1.33153106367843797436547517728, 3.11210321750689923638245189346, 3.49839209218241177546766190160, 4.71883769099416620556295262357, 5.53378037272390217592388190967, 6.10979780039144253905857067493, 7.18481242833963500625169124498, 7.82563842306841597038728593252, 8.523196761884638693677920060893