| L(s) = 1 | + 1.68i·5-s + 3.53i·11-s − 2.93i·13-s − 2.01i·17-s + 1.69·19-s − 1.59i·23-s + 2.16·25-s − 7.94·29-s − 4.95·31-s − 10.4·37-s + 2.86i·41-s − 11.7i·43-s + 6.70·47-s − 2.92·53-s − 5.94·55-s + ⋯ |
| L(s) = 1 | + 0.753i·5-s + 1.06i·11-s − 0.812i·13-s − 0.487i·17-s + 0.389·19-s − 0.333i·23-s + 0.432·25-s − 1.47·29-s − 0.889·31-s − 1.72·37-s + 0.447i·41-s − 1.79i·43-s + 0.977·47-s − 0.401·53-s − 0.802·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8641457761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8641457761\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.68iT - 5T^{2} \) |
| 11 | \( 1 - 3.53iT - 11T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 2.01iT - 17T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 + 1.59iT - 23T^{2} \) |
| 29 | \( 1 + 7.94T + 29T^{2} \) |
| 31 | \( 1 + 4.95T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.86iT - 41T^{2} \) |
| 43 | \( 1 + 11.7iT - 43T^{2} \) |
| 47 | \( 1 - 6.70T + 47T^{2} \) |
| 53 | \( 1 + 2.92T + 53T^{2} \) |
| 59 | \( 1 - 7.75T + 59T^{2} \) |
| 61 | \( 1 + 12.5iT - 61T^{2} \) |
| 67 | \( 1 + 1.70iT - 67T^{2} \) |
| 71 | \( 1 + 6.13iT - 71T^{2} \) |
| 73 | \( 1 + 2.43iT - 73T^{2} \) |
| 79 | \( 1 + 0.865iT - 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 15.9iT - 89T^{2} \) |
| 97 | \( 1 - 9.82iT - 97T^{2} \) |
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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
−7.51747362330311480128072616844, −7.14308039301763593367974839547, −6.55484106970619846914884677235, −5.44215653755733777105094973613, −5.15085304457982027044827844903, −3.99985637318632795677760899067, −3.35962149593466557598343865679, −2.48956122292471012835039954641, −1.68902779472810783489439145444, −0.21690488144836480137114020263,
1.09060526807127929702629432195, 1.88242460281998561249784927129, 3.06512201044111679175307143512, 3.82581863588054564499577206465, 4.50768906161855490240962317375, 5.54860874900433532898334691245, 5.72314358694842832865395086239, 6.83684861003789725611735533451, 7.37765649866352164276365845337, 8.283644476339334759734614644802
