L(s) = 1 | + 4i·7-s − 6i·13-s − 2i·17-s + 4·19-s − 8i·23-s − 6·29-s + 6i·37-s − 10·41-s + 4i·43-s − 8i·47-s − 9·49-s − 10i·53-s + 6·61-s − 4i·67-s − 14i·73-s + ⋯ |
L(s) = 1 | + 1.51i·7-s − 1.66i·13-s − 0.485i·17-s + 0.917·19-s − 1.66i·23-s − 1.11·29-s + 0.986i·37-s − 1.56·41-s + 0.609i·43-s − 1.16i·47-s − 1.28·49-s − 1.37i·53-s + 0.768·61-s − 0.488i·67-s − 1.63i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.444187553\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444187553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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
−8.356117237744262396627248788970, −7.925478168104846975714341066427, −6.86353416069920918335828183447, −6.10896135958146668962288565608, −5.26376993826245499839177109051, −4.99488634805347158017044188817, −3.48334415318213237335190769255, −2.86124222627723501138773372185, −1.98220964954188515704947380266, −0.46173038629256383901340699554,
1.14025669231913413658139872789, 1.97389691906257019638997056139, 3.51830413734142120093643623464, 3.90160661191666830227191516115, 4.77790411426144299987651109889, 5.69531898136205872570559211775, 6.61532021240683204246700409646, 7.37634596758713690071186131137, 7.58645031082790898388984184262, 8.793125230681638335230126352476