| L(s) = 1 | + 2·5-s + 7-s + 4·11-s + 13-s + 6·17-s + 4·19-s − 25-s − 6·29-s + 2·35-s + 6·37-s + 6·41-s + 4·43-s + 49-s + 2·53-s + 8·55-s − 4·59-s − 2·61-s + 2·65-s + 4·67-s − 6·73-s + 4·77-s + 8·79-s − 12·83-s + 12·85-s − 10·89-s + 91-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s + 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.917·19-s − 1/5·25-s − 1.11·29-s + 0.338·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.274·53-s + 1.07·55-s − 0.520·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.702·73-s + 0.455·77-s + 0.900·79-s − 1.31·83-s + 1.30·85-s − 1.05·89-s + 0.104·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.204692505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.204692505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.86808902150910022748272179233, −7.40638260288841524859508132736, −6.48964132982911825714556354626, −5.75789024080702081831307579096, −5.44384505756742781933018364260, −4.32328205164357812303218546295, −3.64183458737992114265619645944, −2.72073979031890227895774674961, −1.65031570417369949078072999027, −1.03907702587109094101826337827,
1.03907702587109094101826337827, 1.65031570417369949078072999027, 2.72073979031890227895774674961, 3.64183458737992114265619645944, 4.32328205164357812303218546295, 5.44384505756742781933018364260, 5.75789024080702081831307579096, 6.48964132982911825714556354626, 7.40638260288841524859508132736, 7.86808902150910022748272179233
