| L(s) = 1 | + i·2-s + 4-s + 3i·8-s + 3·9-s + 4·11-s − 16-s + 3i·18-s + 4i·22-s − 8i·23-s − 2·29-s + 5i·32-s + 3·36-s − 6i·37-s + 12i·43-s + 4·44-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.5·4-s + 1.06i·8-s + 9-s + 1.20·11-s − 0.250·16-s + 0.707i·18-s + 0.852i·22-s − 1.66i·23-s − 0.371·29-s + 0.883i·32-s + 0.5·36-s − 0.986i·37-s + 1.82i·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287525259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287525259\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| 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
−9.771250188868561795391688243002, −8.952772387457675256535099877117, −8.079528927219190463262967036019, −7.24030895518800721395748600639, −6.60973483690631765570905823420, −5.97896574798438697490533614779, −4.77427583873579081767954459657, −3.94471686331882774899431803123, −2.55727884186039395969986671853, −1.36321925300395650269465102503,
1.20481275212509939452365056019, 2.02495550452506870676685756872, 3.47185485966901585591656140513, 3.99216095553573431735843439729, 5.27964669881712485466281931799, 6.46895163930502220218130958741, 7.00666616410248295909397260561, 7.83316777882651927873725804425, 9.088080049500239296415354148394, 9.724662915621575717143485544974
