| L(s) = 1 | − 1.30·2-s − 0.302·4-s + 5-s + 3·8-s − 1.30·10-s + 2.60·11-s + 0.605·13-s − 3.30·16-s − 5.60·17-s + 3.60·19-s − 0.302·20-s − 3.39·22-s + 3·23-s + 25-s − 0.788·26-s − 8.60·29-s − 1.60·31-s − 1.69·32-s + 7.30·34-s + 2·37-s − 4.69·38-s + 3·40-s + 2.60·41-s − 6.60·43-s − 0.788·44-s − 3.90·46-s + 5.21·47-s + ⋯ |
| L(s) = 1 | − 0.921·2-s − 0.151·4-s + 0.447·5-s + 1.06·8-s − 0.411·10-s + 0.785·11-s + 0.167·13-s − 0.825·16-s − 1.35·17-s + 0.827·19-s − 0.0677·20-s − 0.723·22-s + 0.625·23-s + 0.200·25-s − 0.154·26-s − 1.59·29-s − 0.288·31-s − 0.300·32-s + 1.25·34-s + 0.328·37-s − 0.761·38-s + 0.474·40-s + 0.406·41-s − 1.00·43-s − 0.118·44-s − 0.576·46-s + 0.760·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 - 0.605T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 + 1.60T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 5.60T + 53T^{2} \) |
| 59 | \( 1 + 8.60T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 + 4.39T + 79T^{2} \) |
| 83 | \( 1 - 3T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.57094072372789917586161053463, −7.22825121128422490467238396581, −6.33587450953180132087987062471, −5.60779071653156717468587005586, −4.69797011112399225276934898878, −4.10296677826639922753420199125, −3.09522549699185941018816894422, −1.93110911649560007386848822742, −1.24938231494982995263418162487, 0,
1.24938231494982995263418162487, 1.93110911649560007386848822742, 3.09522549699185941018816894422, 4.10296677826639922753420199125, 4.69797011112399225276934898878, 5.60779071653156717468587005586, 6.33587450953180132087987062471, 7.22825121128422490467238396581, 7.57094072372789917586161053463
