| L(s) = 1 | − 2.82i·5-s + i·7-s − 0.635·11-s + 0.550·13-s + 3.14i·17-s − 4i·19-s − 3.14·23-s − 3.00·25-s − 5.19i·29-s + 10.7i·31-s + 2.82·35-s − 8.89·37-s − 3.46i·41-s + 0.550i·43-s − 1.27·47-s + ⋯ |
| L(s) = 1 | − 1.26i·5-s + 0.377i·7-s − 0.191·11-s + 0.152·13-s + 0.763i·17-s − 0.917i·19-s − 0.656·23-s − 0.600·25-s − 0.964i·29-s + 1.93i·31-s + 0.478·35-s − 1.46·37-s − 0.541i·41-s + 0.0839i·43-s − 0.185·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1334472370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1334472370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 11 | \( 1 + 0.635T + 11T^{2} \) |
| 13 | \( 1 - 0.550T + 13T^{2} \) |
| 17 | \( 1 - 3.14iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 3.14T + 23T^{2} \) |
| 29 | \( 1 + 5.19iT - 29T^{2} \) |
| 31 | \( 1 - 10.7iT - 31T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 0.550iT - 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + 2.36iT - 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 - 1.44iT - 67T^{2} \) |
| 71 | \( 1 + 5.97T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 13.5iT - 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
| 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
−8.567922951354392846912268417597, −7.78730564175734928008483656474, −6.89996571843766146265771554781, −6.13665928417754459803789585993, −5.34024186276845388570418885365, −4.84751607310683114614116511003, −4.06343459269773291299597157871, −3.14809806138303908064659059353, −2.05055950807791847734200226640, −1.21351447577162771640250664648,
0.03355091358480577318533867946, 1.56989622863231988247112996275, 2.55390699054728251058882411503, 3.31066993989103792816737376624, 3.97385482327176984325246447442, 4.92105606550115619029238102856, 5.85139992381938108638174572630, 6.40616151908556384523915189082, 7.19326343341518253331664898295, 7.62723815195480359274051229138
