L(s) = 1 | + (0.866 + 0.5i)4-s + (0.5 + 0.866i)7-s + (−0.866 − 0.5i)13-s + (0.499 + 0.866i)16-s + (−0.366 − 0.366i)19-s + (−0.866 + 0.5i)25-s + 0.999i·28-s + (0.366 − 1.36i)31-s + (−0.5 − 0.133i)37-s + (1.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + (−0.499 − 0.866i)52-s + 1.73·61-s + 0.999i·64-s + (−1 − i)67-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)4-s + (0.5 + 0.866i)7-s + (−0.866 − 0.5i)13-s + (0.499 + 0.866i)16-s + (−0.366 − 0.366i)19-s + (−0.866 + 0.5i)25-s + 0.999i·28-s + (0.366 − 1.36i)31-s + (−0.5 − 0.133i)37-s + (1.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + (−0.499 − 0.866i)52-s + 1.73·61-s + 0.999i·64-s + (−1 − i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.185739203\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.185739203\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 5 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 - 1.73T + T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
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
−10.66829304722149074251141703441, −9.677000167191598151991163238460, −8.729992692397280035212316987105, −7.86298234241593969977573841782, −7.25704756397129989719478031650, −6.12184918493857249235578019908, −5.36327450569760800441968400123, −4.11377982116627186934721578177, −2.80366753766539992535523215536, −2.02337167280871345113185974527,
1.45823378368166458461027255327, 2.62153557837044334587783572346, 4.04986296281894392663774182877, 5.02020567025159185866521503885, 6.08236340971301555016360185405, 7.00413122506102861104146279792, 7.57247009896421151528779520251, 8.596045092680636548029194366197, 9.851412206101690271138308278140, 10.30880754041816921736122443143