| L(s) = 1 | − 2·5-s + 0.208·7-s + 11-s − 13-s + 4·17-s + 1.20·19-s − 0.208·23-s − 25-s + 3.58·29-s − 0.417·35-s + 4·37-s − 1.79·41-s − 7.16·43-s − 3.58·47-s − 6.95·49-s − 4.37·53-s − 2·55-s + 5.58·59-s + 2.41·61-s + 2·65-s + 11.1·67-s − 10.7·73-s + 0.208·77-s + 12.7·79-s − 0.626·83-s − 8·85-s − 9.58·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.0788·7-s + 0.301·11-s − 0.277·13-s + 0.970·17-s + 0.277·19-s − 0.0435·23-s − 0.200·25-s + 0.665·29-s − 0.0705·35-s + 0.657·37-s − 0.279·41-s − 1.09·43-s − 0.522·47-s − 0.993·49-s − 0.600·53-s − 0.269·55-s + 0.726·59-s + 0.309·61-s + 0.248·65-s + 1.36·67-s − 1.26·73-s + 0.0237·77-s + 1.43·79-s − 0.0687·83-s − 0.867·85-s − 1.01·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.497804814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.497804814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 0.208T + 7T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 0.208T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 0.626T + 83T^{2} \) |
| 89 | \( 1 + 9.58T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.113181088173341720275450694882, −7.62211114239343718609574014605, −6.86276035017413460052287765616, −6.11434579549022253659304800456, −5.19653771433609931550570982628, −4.52967773488939178172120354608, −3.64670739331414202002080854247, −3.06270798286681102927528040261, −1.83437768797558287720751433260, −0.66966202704952433993836743030,
0.66966202704952433993836743030, 1.83437768797558287720751433260, 3.06270798286681102927528040261, 3.64670739331414202002080854247, 4.52967773488939178172120354608, 5.19653771433609931550570982628, 6.11434579549022253659304800456, 6.86276035017413460052287765616, 7.62211114239343718609574014605, 8.113181088173341720275450694882
