| L(s) = 1 | + (−0.144 + 1.40i)2-s + (−1.95 − 0.406i)4-s + 3.62i·7-s + (0.855 − 2.69i)8-s − 6.20i·11-s + 0.578·13-s + (−5.10 − 0.524i)14-s + (3.66 + 1.59i)16-s + 1.42i·17-s + 5.62i·19-s + (8.72 + 0.897i)22-s + 5.62i·23-s + (−0.0836 + 0.813i)26-s + (1.47 − 7.10i)28-s + 2i·29-s + ⋯ |
| L(s) = 1 | + (−0.102 + 0.994i)2-s + (−0.979 − 0.203i)4-s + 1.37i·7-s + (0.302 − 0.953i)8-s − 1.87i·11-s + 0.160·13-s + (−1.36 − 0.140i)14-s + (0.917 + 0.398i)16-s + 0.344i·17-s + 1.29i·19-s + (1.86 + 0.191i)22-s + 1.17i·23-s + (−0.0163 + 0.159i)26-s + (0.278 − 1.34i)28-s + 0.371i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8589815595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8589815595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.144 - 1.40i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.62iT - 7T^{2} \) |
| 11 | \( 1 + 6.20iT - 11T^{2} \) |
| 13 | \( 1 - 0.578T + 13T^{2} \) |
| 17 | \( 1 - 1.42iT - 17T^{2} \) |
| 19 | \( 1 - 5.62iT - 19T^{2} \) |
| 23 | \( 1 - 5.62iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 - 6.78iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 2.20iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 8.41T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 5.42T + 79T^{2} \) |
| 83 | \( 1 - 3.25T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 4.84iT - 97T^{2} \) |
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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
−9.307467976116196955588187406085, −8.712216072952752583993060313406, −8.269265180665639233698982951492, −7.40223238478682946553372804322, −6.22177316056187772726618863922, −5.76778552141371408199041861638, −5.28674916206172513533939061675, −3.86192699182929882436636275093, −3.10432955539220638841002163203, −1.45219565969729934609251999291,
0.34629035325498985481628647471, 1.66804062318493638338595568405, 2.67106042189933688841287302826, 3.89410444073049710136452269307, 4.50236851555938723631423883177, 5.16790788137823013543994239092, 6.82800712711113309486754139064, 7.21627497532892109375293975597, 8.176481337836981526339761101914, 9.127429663613407875458758782811
