L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + (−1 − 1.73i)11-s + 0.999i·12-s + (−0.866 − 0.5i)13-s + (−0.499 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 0.999·21-s − i·27-s + (0.866 − 0.499i)28-s + (−0.5 − 0.866i)29-s + (1.73 + 0.999i)33-s + 0.999·39-s − 1.99·44-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s + (−1 − 1.73i)11-s + 0.999i·12-s + (−0.866 − 0.5i)13-s + (−0.499 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 0.999·21-s − i·27-s + (0.866 − 0.499i)28-s + (−0.5 − 0.866i)29-s + (1.73 + 0.999i)33-s + 0.999·39-s − 1.99·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6704937765\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6704937765\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + iT - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + iT - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
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
−9.068537067573258138353627830150, −8.154649239148746612697417257842, −7.51281916147336628990012816517, −6.25601554749837038134915761481, −5.73578291121500193640582110817, −5.15612713794590066002614892395, −4.58730210379015798862283747883, −2.93546368005297230668737447746, −2.15480145750643325413042021160, −0.47057905551869666529201227604,
1.74844549364917717735015847049, 2.43588286705800768065709256296, 3.89452255165608766456183696510, 4.72149918402846750443028530301, 5.38664482695406827814519204075, 6.71836303959055836788253583319, 7.04157838008125457535950302515, 7.64626415017672352578646353459, 8.429598083223345591552211006573, 9.432350606389113097314669894441