L(s) = 1 | − i·3-s + 2·4-s + 2·9-s − 3·11-s − 2i·12-s − 5i·13-s + 4·16-s + 3i·17-s + 2·19-s − 6i·23-s − 5i·27-s − 3·29-s + 4·31-s + 3i·33-s + 4·36-s − 2i·37-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 4-s + 0.666·9-s − 0.904·11-s − 0.577i·12-s − 1.38i·13-s + 16-s + 0.727i·17-s + 0.458·19-s − 1.25i·23-s − 0.962i·27-s − 0.557·29-s + 0.718·31-s + 0.522i·33-s + 0.666·36-s − 0.328i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.138646328\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.138646328\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + iT - 97T^{2} \) |
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
−9.809012076975704178050638223042, −8.492023814308683359530528321013, −7.65212341123739983301926105703, −7.36200319427344189562770520535, −6.20114581194712439796224933213, −5.67177085070311137272905068414, −4.38994178042530406142810336985, −3.05243238467350390075692737071, −2.26000522287000529379967910861, −0.958770528599014897711384588392,
1.52293871530131897144994127778, 2.64881610092155115619941036006, 3.71546315800725028546040145024, 4.74508080399287726146407319234, 5.61613901951469594840923339883, 6.69932945099849126946627826212, 7.32848234581028406475988253106, 8.066021885086306340485902760396, 9.443343083244125355106283534189, 9.722895826047067835675412783330