L(s) = 1 | + i·3-s + i·7-s − 11-s + i·13-s − i·17-s − 21-s + i·27-s + 29-s − i·33-s − 39-s − i·47-s − 49-s + 51-s + 2·71-s − 2i·73-s + ⋯ |
L(s) = 1 | + i·3-s + i·7-s − 11-s + i·13-s − i·17-s − 21-s + i·27-s + 29-s − i·33-s − 39-s − i·47-s − 49-s + 51-s + 2·71-s − 2i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9140492637\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9140492637\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - iT - T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( 1 + 2iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−10.76035020396532464080243620846, −9.931782335995926821818685202912, −9.270719671494445086390622554280, −8.541282495342680240225583349039, −7.42766257220495881120390361797, −6.35368728470318625647203523095, −5.17433253965694296237475362312, −4.67109932167493845430517850771, −3.36215374059236355005664948042, −2.22972192861995293180341740539,
1.09331339495056724340748796325, 2.54749219910442087433699752843, 3.85203172566153858658551399481, 5.05183757213150740405063828488, 6.18675533791111212741284877628, 7.01449192328764480481317833813, 7.891889877853086117941671628195, 8.254657621720474302069963053615, 9.824947976513585554972070817681, 10.46066348006947616896795504313