L(s) = 1 | + i·2-s − 4-s − i·8-s − 3i·11-s − 3.46i·13-s + 16-s + 3·22-s − 5·25-s + 3.46·26-s + 9i·29-s + 1.73i·31-s + i·32-s − 8·37-s − 10.3·41-s + 4·43-s + 3i·44-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.353i·8-s − 0.904i·11-s − 0.960i·13-s + 0.250·16-s + 0.639·22-s − 25-s + 0.679·26-s + 1.67i·29-s + 0.311i·31-s + 0.176i·32-s − 1.31·37-s − 1.62·41-s + 0.609·43-s + 0.452i·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1975822747\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1975822747\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + 5.19iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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
−8.455969567520414395044926347605, −7.896072907301417357961991914892, −7.02695926526481741750041847852, −6.30465943728612215028612105204, −5.46566330435062002101903425304, −4.94986080820336098380206100466, −3.67104615046743863423325712964, −3.08110597321353337030747638761, −1.51021356338277789901656331606, −0.06293047766237565424761538857,
1.63367412647309236589960454103, 2.30404124844821287007999738350, 3.53444700466664810478895880049, 4.29607892244510893422977292974, 5.00795350201739234840604597492, 6.04172122608279827931198105615, 6.87655424135954832313120817881, 7.70086452610952035571031794577, 8.491619743641865904957804874057, 9.322823317434413208374960143745