L(s) = 1 | + (0.5 − 0.866i)7-s + (1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s − 1.73i·31-s + (1 − 1.73i)37-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)49-s + 1.73i·61-s + 2·67-s + (−1.5 + 0.866i)73-s + 2·79-s + (−1.5 + 0.866i)97-s + (−0.5 − 0.866i)109-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)7-s + (1.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s − 1.73i·31-s + (1 − 1.73i)37-s + (−0.5 − 0.866i)43-s + (−0.499 − 0.866i)49-s + 1.73i·61-s + 2·67-s + (−1.5 + 0.866i)73-s + 2·79-s + (−1.5 + 0.866i)97-s + (−0.5 − 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.284538936\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.284538936\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.73iT - T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
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
−9.323767556772386123464650195844, −8.183390758650585172172434113875, −7.59959335662148466210898760268, −7.06708291030674480868014192094, −5.86504623954550945699731926177, −5.29885318682443581622708509130, −4.14460132046167396445146082816, −3.60346386295415505963506343384, −2.26393279897665724332223860809, −1.06753734976265999161598554399,
1.34889658328521254773296566407, 2.57558457321869197982958817010, 3.35829728128863022803359499921, 4.73595378358401665628286954695, 5.14561318615506452647437531644, 6.17088919429311617222424859917, 6.88783576059875584445743675222, 7.926850084592311896253901883400, 8.390185368178959278672296847582, 9.332015813663208421338619012778