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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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
−8.731600469424218256517314059154, −7.968546495743276285892234566496, −7.05202789774082241386866214904, −6.21648940658876017122598339493, −5.52302718317988151036413742129, −4.72316216292546896078046326765, −3.57776422736899308372712736248, −2.74069323824921438756885998149, −1.92578464284909834771248956834, 0,
1.37374890790367982153748551298, 2.33559924612471592659008956244, 3.93048013535419094775398629864, 4.48307130811217325809175245918, 4.86329120212920322314284534981, 6.18837101056719285953327028322, 7.13486764868327116963586724563, 7.55533611077282287995279534489, 8.488638825465717007412615424399