| L(s) = 1 | + 0.206·5-s + 3.03i·7-s − 5.02i·11-s − 4.03i·13-s + 4.81i·17-s − 1.03·19-s + 5.43·23-s − 4.95·25-s + 2.90·29-s − 1.23i·31-s + 0.629i·35-s − 8.77i·37-s − 0.979i·41-s + 3.18·43-s − 3.26·47-s + ⋯ |
| L(s) = 1 | + 0.0925·5-s + 1.14i·7-s − 1.51i·11-s − 1.12i·13-s + 1.16i·17-s − 0.238·19-s + 1.13·23-s − 0.991·25-s + 0.539·29-s − 0.222i·31-s + 0.106i·35-s − 1.44i·37-s − 0.152i·41-s + 0.485·43-s − 0.476·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.488891499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.488891499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.206T + 5T^{2} \) |
| 7 | \( 1 - 3.03iT - 7T^{2} \) |
| 11 | \( 1 + 5.02iT - 11T^{2} \) |
| 13 | \( 1 + 4.03iT - 13T^{2} \) |
| 17 | \( 1 - 4.81iT - 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 5.43T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 + 1.23iT - 31T^{2} \) |
| 37 | \( 1 + 8.77iT - 37T^{2} \) |
| 41 | \( 1 + 0.979iT - 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 + 3.26T + 47T^{2} \) |
| 53 | \( 1 + 7.91T + 53T^{2} \) |
| 59 | \( 1 - 14.8iT - 59T^{2} \) |
| 61 | \( 1 + 9.53iT - 61T^{2} \) |
| 67 | \( 1 - 0.735T + 67T^{2} \) |
| 71 | \( 1 - 2.33T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 14.1iT - 79T^{2} \) |
| 83 | \( 1 + 4.23iT - 83T^{2} \) |
| 89 | \( 1 + 11.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−8.200758706587157469362382046364, −7.50743003596484787220068293687, −6.36167824056847936774678635454, −5.76453248035444798797482606778, −5.51239792755613298501970980430, −4.34102266867203352681141590266, −3.33278110736200050120796630266, −2.79909255935212660584547717269, −1.71897256245016825940984367244, −0.42777993559747970056986657984,
1.08758120490554610050782233685, 2.01769411257238888089809725724, 3.02178837486166528212812209830, 4.11150544414637690793059856174, 4.59076064707113434356836921370, 5.23129759458219933185133820715, 6.62515879075996061842506133179, 6.83024059440550196867557193004, 7.51429700573703427952442628807, 8.226555627726181775361021350146
