L(s) = 1 | − 1.73i·5-s − 1.73i·11-s − 5.19i·17-s + 7·19-s − 8.66i·23-s + 2.00·25-s + 6·29-s + 5·31-s − 5·37-s + 6.92i·41-s + 3.46i·43-s − 3·47-s + 9·53-s − 2.99·55-s − 9·59-s + ⋯ |
L(s) = 1 | − 0.774i·5-s − 0.522i·11-s − 1.26i·17-s + 1.60·19-s − 1.80i·23-s + 0.400·25-s + 1.11·29-s + 0.898·31-s − 0.821·37-s + 1.08i·41-s + 0.528i·43-s − 0.437·47-s + 1.23·53-s − 0.404·55-s − 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.039602612\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.039602612\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 1.73iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 8.66iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 9T + 59T^{2} \) |
| 61 | \( 1 - 8.66iT - 61T^{2} \) |
| 67 | \( 1 + 5.19iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 + 5.19iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 12.1iT - 89T^{2} \) |
| 97 | \( 1 + 6.92iT - 97T^{2} \) |
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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
−7.83781235944996863053626419440, −7.01526660988118157663818510466, −6.39912989782233431593518184315, −5.51212847533016117050883062159, −4.83535797889663773223953186474, −4.40825443130862067097992602740, −3.11210540632863737282327953495, −2.70571482846972073937925298183, −1.24647579997483775547554209837, −0.58340925005011158905411957682,
1.14964101511043087545773266455, 2.06258198811848863366898403462, 3.13158175221822986614682777890, 3.57906902433984019215726583318, 4.58109066689842512292964496460, 5.40806085644678978779470107830, 6.01268614191039178213110261038, 6.94454934441386724143771326927, 7.26658465646809936746860947552, 8.068050636733633024146880881022