| L(s) = 1 | − 3.16·3-s + 2.23i·5-s + 4.24i·7-s + 7.00·9-s − 7.07i·15-s − 13.4i·21-s + 1.41i·23-s − 5.00·25-s − 12.6·27-s + 8.94i·29-s − 9.48·35-s − 12·41-s − 3.16·43-s + 15.6i·45-s − 9.89i·47-s + ⋯ |
| L(s) = 1 | − 1.82·3-s + 0.999i·5-s + 1.60i·7-s + 2.33·9-s − 1.82i·15-s − 2.92i·21-s + 0.294i·23-s − 1.00·25-s − 2.43·27-s + 1.66i·29-s − 1.60·35-s − 1.87·41-s − 0.482·43-s + 2.33i·45-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(-0.442641i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.442641i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 8.94iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 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
−11.15300320442692612992440345658, −10.35837642340382558450938013590, −9.578360623351604598425923251396, −8.379369682509881183453623199089, −7.00311258424117998155116840347, −6.49132396223790901103847816349, −5.54470320117646251012923532883, −5.05152438700207788296463899958, −3.43068112290929020497867994407, −1.91303189722742708556565540334,
0.32930517265278970851810051168, 1.34868851682861573660075126866, 4.02881501329376439029045034743, 4.59935673512323901029284567037, 5.52033782735803299521553417665, 6.46460209123470672769002340634, 7.26716864061761554467849436925, 8.219074314540231395010763315854, 9.728509926877035802461870515902, 10.20346304680597843853554148870
