| L(s) = 1 | + 3.04·5-s + 0.418i·7-s − 1.27i·11-s + 3.88·17-s − 4.35i·19-s − 4i·23-s + 4.27·25-s + 1.27i·35-s − 5.67i·43-s − 2.72i·47-s + 6.82·49-s − 3.88i·55-s + 11.2·61-s + 5.82·73-s + 0.533·77-s + ⋯ |
| L(s) = 1 | + 1.36·5-s + 0.158i·7-s − 0.384i·11-s + 0.941·17-s − 0.999i·19-s − 0.834i·23-s + 0.854·25-s + 0.215i·35-s − 0.865i·43-s − 0.397i·47-s + 0.974·49-s − 0.523i·55-s + 1.44·61-s + 0.681·73-s + 0.0608·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.453951824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453951824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 - 3.04T + 5T^{2} \) |
| 7 | \( 1 - 0.418iT - 7T^{2} \) |
| 11 | \( 1 + 1.27iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 5.67iT - 43T^{2} \) |
| 47 | \( 1 + 2.72iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 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
−8.859495706170006530808379828996, −8.145978724543138733022006738265, −7.07385678094807746391383126991, −6.43895461777957136857081064609, −5.58754167740281668348539588585, −5.15112122397630590843068923741, −3.96452954975559293949829375522, −2.83337342591387329049945736514, −2.10768809178744546692230187618, −0.877125570191963750417675690111,
1.24268684183096868876871750484, 2.06447149317107247708074581218, 3.13183454999840182586927457066, 4.12201545468196995989059584951, 5.24982617603440411949726488503, 5.74315928240599461809969983900, 6.47786487050999726629329870475, 7.39113456691437158424534004688, 8.088204034873065254968631797852, 9.054877143862837238411646340852
