L(s) = 1 | + (−0.264 + 0.457i)2-s + (1.72 − 0.197i)3-s + (0.860 + 1.48i)4-s + (0.5 + 0.866i)5-s + (−0.364 + 0.840i)6-s + (−1.64 + 2.85i)7-s − 1.96·8-s + (2.92 − 0.680i)9-s − 0.528·10-s + (0.276 − 0.478i)11-s + (1.77 + 2.39i)12-s + (−0.5 − 0.866i)13-s + (−0.872 − 1.51i)14-s + (1.03 + 1.39i)15-s + (−1.20 + 2.07i)16-s + 6.44·17-s + ⋯ |
L(s) = 1 | + (−0.186 + 0.323i)2-s + (0.993 − 0.114i)3-s + (0.430 + 0.744i)4-s + (0.223 + 0.387i)5-s + (−0.148 + 0.343i)6-s + (−0.623 + 1.07i)7-s − 0.695·8-s + (0.973 − 0.226i)9-s − 0.167·10-s + (0.0832 − 0.144i)11-s + (0.512 + 0.690i)12-s + (−0.138 − 0.240i)13-s + (−0.233 − 0.403i)14-s + (0.266 + 0.359i)15-s + (−0.300 + 0.519i)16-s + 1.56·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0542 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0542 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33608 + 1.41064i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33608 + 1.41064i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 + 0.197i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.264 - 0.457i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (1.64 - 2.85i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.276 + 0.478i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 + 7.62T + 19T^{2} \) |
| 23 | \( 1 + (-1.05 - 1.83i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.658 - 1.14i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.07 + 7.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + (-4.67 - 8.09i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.37 + 5.84i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.84 + 4.93i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4.88T + 53T^{2} \) |
| 59 | \( 1 + (0.860 + 1.49i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.714 - 1.23i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.567 + 0.982i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.65T + 71T^{2} \) |
| 73 | \( 1 - 3.42T + 73T^{2} \) |
| 79 | \( 1 + (5.38 - 9.32i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.45 + 9.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 8.64T + 89T^{2} \) |
| 97 | \( 1 + (3.18 - 5.52i)T + (-48.5 - 84.0i)T^{2} \) |
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
−10.86732444493105783598886128589, −9.694920887311162825178980488672, −9.121114456150103553715317022549, −8.180908318622159887811497994219, −7.58372745217746141289944170675, −6.51537165304422964580497184912, −5.78061150576778027569684528985, −3.93867467272521802583227608184, −2.97873890976396981161998809499, −2.23251806280280058082418327306,
1.09588105869568455336104962579, 2.37615795268384936509687222742, 3.59823058228249351367276210135, 4.64682436327419848810718148883, 6.04111479697296424442774668968, 6.96150270975161771210239608403, 7.84595959187783899257245415644, 9.007622562534178897005073253909, 9.653278104532633994548626834835, 10.36975251128535074667584991484