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 | \( 1 + (-5.62 - 5.62i)T + 97iT^{2} \) |
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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
−10.03211189868178768905774824569, −9.335242022997687818943784975145, −8.580589847273714236608530192441, −7.68078039733278760877619036309, −7.13365951105109447909103539168, −5.48224390573806216737916671233, −4.57993356332144637607605883815, −3.61351098797431580746355261257, −2.60981636861754704071151764954, −1.45285797262224882716125973356,
1.74770689437084157650291853134, 3.08522018851300574621899860915, 3.95623454297946745081491373742, 5.17990249381614560963710770811, 6.03517787774143039857338374332, 7.04529364156974005761340892863, 7.72029880267550769142015632215, 8.778158328329351060298796922854, 9.883204134056032940130038863161, 10.31127340038399318820090690131