| L(s) = 1 | + i·3-s − 1.41i·5-s + 2.82·7-s − 9-s − 4.24i·13-s + 1.41·15-s + 4·17-s + 8i·19-s + 2.82i·21-s − 5.65·23-s + 2.99·25-s − i·27-s − 1.41i·29-s + 2.82·31-s − 4.00i·35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.632i·5-s + 1.06·7-s − 0.333·9-s − 1.17i·13-s + 0.365·15-s + 0.970·17-s + 1.83i·19-s + 0.617i·21-s − 1.17·23-s + 0.599·25-s − 0.192i·27-s − 0.262i·29-s + 0.508·31-s − 0.676i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.163608702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.163608702\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 2.82T + 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 1.41iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
| 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
−8.361315744271873036466220710439, −8.209810997609508974396450582498, −7.55308516843786072076827483853, −6.14015111696037477604273799467, −5.52395424823493957517575883437, −4.91314539657881683093716580955, −4.04847040347178759116119016932, −3.25404219431230265828656767223, −1.96051687333810829985279519272, −0.885387032640070763273595811151,
0.985891028854160328539798220229, 2.09967088358890291444175536747, 2.82886453843289754787361789768, 4.08271455230211738526706959628, 4.81272973120372514942858512626, 5.76225594450571301618011456935, 6.55546295380232342015329326449, 7.33166908343161300982807365495, 7.70723085948363241404064973401, 8.780424750891438062768136178726
