| 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 | \( 1 - 97T^{2} \) |
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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
−9.724662915621575717143485544974, −9.088080049500239296415354148394, −7.83316777882651927873725804425, −7.00666616410248295909397260561, −6.46895163930502220218130958741, −5.27964669881712485466281931799, −3.99216095553573431735843439729, −3.47185485966901585591656140513, −2.02495550452506870676685756872, −1.20481275212509939452365056019,
1.36321925300395650269465102503, 2.55727884186039395969986671853, 3.94471686331882774899431803123, 4.77427583873579081767954459657, 5.97896574798438697490533614779, 6.60973483690631765570905823420, 7.24030895518800721395748600639, 8.079528927219190463262967036019, 8.952772387457675256535099877117, 9.771250188868561795391688243002
