L(s) = 1 | + (−0.5 + 0.866i)4-s + 7-s + (1.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)28-s − 1.73i·31-s + 1.73i·37-s + (0.5 + 0.866i)43-s + (−1.5 + 0.866i)52-s + (−0.5 + 0.866i)61-s + 0.999·64-s + (−1.5 − 0.866i)67-s + (0.5 + 0.866i)73-s + (1.5 − 0.866i)79-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)4-s + 7-s + (1.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s + (0.5 − 0.866i)25-s + (−0.5 + 0.866i)28-s − 1.73i·31-s + 1.73i·37-s + (0.5 + 0.866i)43-s + (−1.5 + 0.866i)52-s + (−0.5 + 0.866i)61-s + 0.999·64-s + (−1.5 − 0.866i)67-s + (0.5 + 0.866i)73-s + (1.5 − 0.866i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.340253485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.340253485\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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
−8.840072184102666070232218288755, −8.115951285452561963505383518771, −7.77305045492050339229067379153, −6.66969113749752101259928434276, −6.00817633412880593672152776205, −4.81180736765956577315101229411, −4.33931518699122910180995037902, −3.54018624267618144778190584879, −2.46967940259246759521080510718, −1.27923637187373894372943988321,
1.02962987883543418812191685697, 1.83043379647876163264005803160, 3.27224063039232803522736910679, 4.13738257279103385877408229951, 5.09407706400131017447352677758, 5.53478311431452852529617122078, 6.33087742920474596172874830226, 7.28020870363842873542245277028, 8.164575142244402909451989518430, 8.773531421980081763754388289357