L(s) = 1 | + 9·2-s + 49·4-s − 24·5-s + 72·7-s + 153·8-s − 216·10-s + 306·13-s + 648·14-s − 191·16-s − 1.20e3·17-s − 774·19-s − 1.17e3·20-s + 4.62e3·23-s − 2.54e3·25-s + 2.75e3·26-s + 3.52e3·28-s + 7.68e3·29-s + 5.42e3·31-s − 6.61e3·32-s − 1.08e4·34-s − 1.72e3·35-s + 3.45e3·37-s − 6.96e3·38-s − 3.67e3·40-s − 7.86e3·41-s + 1.57e4·43-s + 4.16e4·46-s + ⋯ |
L(s) = 1 | + 1.59·2-s + 1.53·4-s − 0.429·5-s + 0.555·7-s + 0.845·8-s − 0.683·10-s + 0.502·13-s + 0.883·14-s − 0.186·16-s − 1.01·17-s − 0.491·19-s − 0.657·20-s + 1.82·23-s − 0.815·25-s + 0.798·26-s + 0.850·28-s + 1.69·29-s + 1.01·31-s − 1.14·32-s − 1.61·34-s − 0.238·35-s + 0.414·37-s − 0.782·38-s − 0.362·40-s − 0.730·41-s + 1.30·43-s + 2.90·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(6.098575235\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.098575235\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 9 T + p^{5} T^{2} \) |
| 5 | \( 1 + 24 T + p^{5} T^{2} \) |
| 7 | \( 1 - 72 T + p^{5} T^{2} \) |
| 13 | \( 1 - 306 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1206 T + p^{5} T^{2} \) |
| 19 | \( 1 + 774 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4626 T + p^{5} T^{2} \) |
| 29 | \( 1 - 7686 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5428 T + p^{5} T^{2} \) |
| 37 | \( 1 - 3454 T + p^{5} T^{2} \) |
| 41 | \( 1 + 7866 T + p^{5} T^{2} \) |
| 43 | \( 1 - 15786 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6402 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21684 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27420 T + p^{5} T^{2} \) |
| 61 | \( 1 - 52866 T + p^{5} T^{2} \) |
| 67 | \( 1 - 25012 T + p^{5} T^{2} \) |
| 71 | \( 1 + 65058 T + p^{5} T^{2} \) |
| 73 | \( 1 + 26676 T + p^{5} T^{2} \) |
| 79 | \( 1 + 18612 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 - 41670 T + p^{5} T^{2} \) |
| 97 | \( 1 - 40694 T + p^{5} T^{2} \) |
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
−8.922846712735330427629237430134, −8.284557773122472605845388302467, −7.09271197197928551945852242094, −6.49381936881124855525513382433, −5.53689559961804616720279564341, −4.63130217811030052396482728389, −4.16181903747777818072799957629, −3.07378063749227953609079328709, −2.22569031009299844403830233043, −0.832662833210570855000195893043,
0.832662833210570855000195893043, 2.22569031009299844403830233043, 3.07378063749227953609079328709, 4.16181903747777818072799957629, 4.63130217811030052396482728389, 5.53689559961804616720279564341, 6.49381936881124855525513382433, 7.09271197197928551945852242094, 8.284557773122472605845388302467, 8.922846712735330427629237430134