L(s) = 1 | + (−1.5 − 0.866i)3-s + (2.13 + 0.656i)5-s + (1.13 − 2.38i)7-s + (2.63 − 4.56i)11-s + 2.62i·13-s + (−2.63 − 2.83i)15-s + (0.362 + 0.209i)17-s + (1.63 + 2.83i)19-s + (−3.77 + 2.59i)21-s + (−6.77 + 3.91i)23-s + (4.13 + 2.80i)25-s + 5.19i·27-s − 4.27·29-s + (−1.63 + 2.83i)31-s + (−7.91 + 4.56i)33-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (0.955 + 0.293i)5-s + (0.429 − 0.902i)7-s + (0.795 − 1.37i)11-s + 0.728i·13-s + (−0.680 − 0.732i)15-s + (0.0879 + 0.0507i)17-s + (0.375 + 0.650i)19-s + (−0.823 + 0.566i)21-s + (−1.41 + 0.815i)23-s + (0.827 + 0.561i)25-s + 0.999i·27-s − 0.793·29-s + (−0.294 + 0.509i)31-s + (−1.37 + 0.795i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.924189 - 0.393938i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.924189 - 0.393938i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.13 - 0.656i)T \) |
| 7 | \( 1 + (-1.13 + 2.38i)T \) |
good | 3 | \( 1 + (1.5 + 0.866i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.63 + 4.56i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 + (-0.362 - 0.209i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 - 2.83i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.77 - 3.91i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 + (1.63 - 2.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.63 + 4.98i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.72T + 41T^{2} \) |
| 43 | \( 1 + 2.15iT - 43T^{2} \) |
| 47 | \( 1 + (5.63 - 3.25i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.91 - 2.83i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.63 - 2.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.77 - 11.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.04 - 1.76i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 + (5.63 + 3.25i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.63 - 6.30i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.40iT - 83T^{2} \) |
| 89 | \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−13.13872424568665578146787660965, −11.75568820183655371573959908688, −11.21843576810834260271690268938, −10.10377558976984511801736746573, −8.947230672484003728126750947123, −7.43183605540843381710365341393, −6.30595027531308705796128842965, −5.63316204669489697555740767505, −3.75951498426771877438300887117, −1.39979249848421277800010502964,
2.15587708548672709365535941029, 4.59456943712169306888869972990, 5.45491614499110740010176072285, 6.43483643276764273708882513124, 8.125364236811436706790750534217, 9.467177760589211512179466068946, 10.08760828683354188672923292406, 11.34300483319928994734301342634, 12.16142123596549995252772394540, 13.11033028946261739829795741961