L(s) = 1 | + (0.664 + 0.664i)2-s + (1.63 + 0.578i)3-s − 1.11i·4-s + (0.459 + 2.18i)5-s + (0.700 + 1.46i)6-s + (2.07 − 2.07i)8-s + (2.33 + 1.88i)9-s + (−1.14 + 1.75i)10-s − 0.727i·11-s + (0.646 − 1.82i)12-s + (−1.44 − 1.44i)13-s + (−0.515 + 3.83i)15-s + 0.515·16-s + (5.19 + 5.19i)17-s + (0.293 + 2.80i)18-s − 0.767i·19-s + ⋯ |
L(s) = 1 | + (0.469 + 0.469i)2-s + (0.942 + 0.334i)3-s − 0.558i·4-s + (0.205 + 0.978i)5-s + (0.285 + 0.599i)6-s + (0.732 − 0.732i)8-s + (0.776 + 0.629i)9-s + (−0.363 + 0.556i)10-s − 0.219i·11-s + (0.186 − 0.526i)12-s + (−0.400 − 0.400i)13-s + (−0.133 + 0.991i)15-s + 0.128·16-s + (1.25 + 1.25i)17-s + (0.0691 + 0.660i)18-s − 0.175i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.58752 + 1.22413i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.58752 + 1.22413i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.63 - 0.578i)T \) |
| 5 | \( 1 + (-0.459 - 2.18i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.664 - 0.664i)T + 2iT^{2} \) |
| 11 | \( 1 + 0.727iT - 11T^{2} \) |
| 13 | \( 1 + (1.44 + 1.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.19 - 5.19i)T + 17iT^{2} \) |
| 19 | \( 1 + 0.767iT - 19T^{2} \) |
| 23 | \( 1 + (-2.29 + 2.29i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 0.419T + 31T^{2} \) |
| 37 | \( 1 + (4.45 - 4.45i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.44iT - 41T^{2} \) |
| 43 | \( 1 + (5.15 + 5.15i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.97 - 4.97i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.85 + 3.85i)T - 53iT^{2} \) |
| 59 | \( 1 - 1.61T + 59T^{2} \) |
| 61 | \( 1 + 9.57T + 61T^{2} \) |
| 67 | \( 1 + (-5.05 + 5.05i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.06iT - 71T^{2} \) |
| 73 | \( 1 + (11.1 + 11.1i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.70iT - 79T^{2} \) |
| 83 | \( 1 + (1.83 - 1.83i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + (-5.62 + 5.62i)T - 97iT^{2} \) |
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
−10.31127340038399318820090690131, −9.883204134056032940130038863161, −8.778158328329351060298796922854, −7.72029880267550769142015632215, −7.04529364156974005761340892863, −6.03517787774143039857338374332, −5.17990249381614560963710770811, −3.95623454297946745081491373742, −3.08522018851300574621899860915, −1.74770689437084157650291853134,
1.45285797262224882716125973356, 2.60981636861754704071151764954, 3.61351098797431580746355261257, 4.57993356332144637607605883815, 5.48224390573806216737916671233, 7.13365951105109447909103539168, 7.68078039733278760877619036309, 8.580589847273714236608530192441, 9.335242022997687818943784975145, 10.03211189868178768905774824569