| 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 | \( 1 + 8.66iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.322823317434413208374960143745, −8.491619743641865904957804874057, −7.70086452610952035571031794577, −6.87655424135954832313120817881, −6.04172122608279827931198105615, −5.00795350201739234840604597492, −4.29607892244510893422977292974, −3.53444700466664810478895880049, −2.30404124844821287007999738350, −1.63367412647309236589960454103,
0.06293047766237565424761538857, 1.51021356338277789901656331606, 3.08110597321353337030747638761, 3.67104615046743863423325712964, 4.94986080820336098380206100466, 5.46566330435062002101903425304, 6.30465943728612215028612105204, 7.02695926526481741750041847852, 7.896072907301417357961991914892, 8.455969567520414395044926347605
