| L(s) = 1 | − 2.23i·5-s − 9.10i·7-s − 10.3i·11-s + 7.61·13-s + 2.71i·17-s + 31.0i·19-s + (18.5 − 13.6i)23-s − 5.00·25-s + 45.1·29-s + 12.3·31-s − 20.3·35-s + 18.1i·37-s + 22.2·41-s + 55.9i·43-s + 82.5·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 1.30i·7-s − 0.937i·11-s + 0.585·13-s + 0.159i·17-s + 1.63i·19-s + (0.805 − 0.593i)23-s − 0.200·25-s + 1.55·29-s + 0.397·31-s − 0.581·35-s + 0.490i·37-s + 0.542·41-s + 1.30i·43-s + 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 + 0.805i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.593 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.515860841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.515860841\) |
| \(L(2)\) |
|
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 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-18.5 + 13.6i)T \) |
| good | 7 | \( 1 + 9.10iT - 49T^{2} \) |
| 11 | \( 1 + 10.3iT - 121T^{2} \) |
| 13 | \( 1 - 7.61T + 169T^{2} \) |
| 17 | \( 1 - 2.71iT - 289T^{2} \) |
| 19 | \( 1 - 31.0iT - 361T^{2} \) |
| 29 | \( 1 - 45.1T + 841T^{2} \) |
| 31 | \( 1 - 12.3T + 961T^{2} \) |
| 37 | \( 1 - 18.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 22.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 55.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 82.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 9.81iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.51T + 3.48e3T^{2} \) |
| 61 | \( 1 - 46.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 63.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 104.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 140.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 3.26iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 82.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.56iT - 9.40e3T^{2} \) |
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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
−8.215134691708303391882063570583, −7.52460373928837936067035007221, −6.56659365971166197912060174478, −6.05977252386614078994553607544, −5.10938850793655667595569788573, −4.21127095490353250109276239064, −3.69204097163323273159756767082, −2.71560633544272934449924429145, −1.22611950643951896523675259754, −0.789211662256234398859350252453,
0.816697775773069623647749980214, 2.22845783632340152559292947891, 2.65444852555247136899832913967, 3.69175102402073485238917213183, 4.78327009415379525513999553970, 5.29833211077073003416947477147, 6.25065239625233386228857832032, 6.84706518430067692494027397549, 7.54720405577817913070587320907, 8.490579679817108075704219772333
