| L(s) = 1 | + 0.908·2-s − 1.17·4-s − 1.84·5-s − 2.88·8-s − 1.67·10-s − 3.73·11-s − 3.52·13-s − 0.274·16-s + 6.63·17-s − 5.54·19-s + 2.16·20-s − 3.39·22-s − 0.379·23-s − 1.59·25-s − 3.20·26-s + 3.54·29-s + 4.54·31-s + 5.52·32-s + 6.03·34-s − 11.6·37-s − 5.03·38-s + 5.32·40-s + 6.30·41-s + 6.46·43-s + 4.38·44-s − 0.344·46-s + 6.14·47-s + ⋯ |
| L(s) = 1 | + 0.642·2-s − 0.586·4-s − 0.824·5-s − 1.01·8-s − 0.530·10-s − 1.12·11-s − 0.977·13-s − 0.0685·16-s + 1.60·17-s − 1.27·19-s + 0.484·20-s − 0.723·22-s − 0.0791·23-s − 0.319·25-s − 0.628·26-s + 0.659·29-s + 0.816·31-s + 0.975·32-s + 1.03·34-s − 1.91·37-s − 0.817·38-s + 0.841·40-s + 0.984·41-s + 0.985·43-s + 0.660·44-s − 0.0508·46-s + 0.895·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.046151079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.046151079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.908T + 2T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 - 6.63T + 17T^{2} \) |
| 19 | \( 1 + 5.54T + 19T^{2} \) |
| 23 | \( 1 + 0.379T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 6.30T + 41T^{2} \) |
| 43 | \( 1 - 6.46T + 43T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 + 0.224T + 53T^{2} \) |
| 59 | \( 1 - 0.866T + 59T^{2} \) |
| 61 | \( 1 + 0.566T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 0.137T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 - 2.84T + 97T^{2} \) |
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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
−8.542227401834278015070053300515, −7.921398732879034177495470504542, −7.38210654626032307905081035192, −6.24480286153900029663458962115, −5.43234400295142428817399131835, −4.81387283066382277306445423390, −4.06661920643688059558300023828, −3.25867475828440209263503282557, −2.39076234541123587998629583580, −0.54021621137283485014476902286,
0.54021621137283485014476902286, 2.39076234541123587998629583580, 3.25867475828440209263503282557, 4.06661920643688059558300023828, 4.81387283066382277306445423390, 5.43234400295142428817399131835, 6.24480286153900029663458962115, 7.38210654626032307905081035192, 7.921398732879034177495470504542, 8.542227401834278015070053300515
