| L(s) = 1 | + 1.41i·5-s + 2·7-s + 1.41i·11-s − 6.32i·13-s + 4.24i·17-s + (3 + 3.16i)19-s − 7.07i·23-s + 2.99·25-s + 4.47·29-s − 6.32i·31-s + 2.82i·35-s − 4.47·41-s − 4·43-s + 7.07i·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 0.632i·5-s + 0.755·7-s + 0.426i·11-s − 1.75i·13-s + 1.02i·17-s + (0.688 + 0.725i)19-s − 1.47i·23-s + 0.599·25-s + 0.830·29-s − 1.13i·31-s + 0.478i·35-s − 0.698·41-s − 0.609·43-s + 1.03i·47-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.083082335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.083082335\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3 - 3.16i)T \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 6.32iT - 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 23 | \( 1 + 7.07iT - 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 6.32iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 1.41iT - 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 6.32iT - 97T^{2} \) |
| 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
−8.426323113509611641150807092384, −8.261973434726807941996670375831, −7.39751250150651082132449033402, −6.56717073769629829640812685244, −5.74101773079112090858776722990, −5.00856520536625212872518152796, −4.04591527062066017000191404609, −3.09018964519109165233542858563, −2.22593475380799848415361888597, −0.913453083315965222952609343829,
0.968845206439867045990154352915, 1.91539942357036517008013627298, 3.12483124892572180294816659802, 4.17701121121912962874466732426, 5.02603679095772538397602354344, 5.38769261356314720497995424846, 6.85154884078563464608360045766, 7.05949027323186535955978590599, 8.246527051769021981962578960969, 8.813263474249672586339569017092
