| L(s) = 1 | + 6·3-s − 118·7-s − 207·9-s − 192·11-s − 1.10e3·13-s − 762·17-s + 2.74e3·19-s − 708·21-s + 1.56e3·23-s − 2.70e3·27-s + 5.91e3·29-s + 6.86e3·31-s − 1.15e3·33-s + 5.51e3·37-s − 6.63e3·39-s − 378·41-s − 2.43e3·43-s + 1.31e4·47-s − 2.88e3·49-s − 4.57e3·51-s + 9.17e3·53-s + 1.64e4·57-s + 3.49e4·59-s − 9.83e3·61-s + 2.44e4·63-s + 3.37e4·67-s + 9.39e3·69-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 0.910·7-s − 0.851·9-s − 0.478·11-s − 1.81·13-s − 0.639·17-s + 1.74·19-s − 0.350·21-s + 0.617·23-s − 0.712·27-s + 1.30·29-s + 1.28·31-s − 0.184·33-s + 0.662·37-s − 0.698·39-s − 0.0351·41-s − 0.200·43-s + 0.866·47-s − 0.171·49-s − 0.246·51-s + 0.448·53-s + 0.670·57-s + 1.30·59-s − 0.338·61-s + 0.775·63-s + 0.917·67-s + 0.237·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.413078436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.413078436\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 118 T + p^{5} T^{2} \) |
| 11 | \( 1 + 192 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1106 T + p^{5} T^{2} \) |
| 17 | \( 1 + 762 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2740 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1566 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5910 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6868 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5518 T + p^{5} T^{2} \) |
| 41 | \( 1 + 378 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2434 T + p^{5} T^{2} \) |
| 47 | \( 1 - 13122 T + p^{5} T^{2} \) |
| 53 | \( 1 - 9174 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34980 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9838 T + p^{5} T^{2} \) |
| 67 | \( 1 - 33722 T + p^{5} T^{2} \) |
| 71 | \( 1 + 70212 T + p^{5} T^{2} \) |
| 73 | \( 1 + 21986 T + p^{5} T^{2} \) |
| 79 | \( 1 + 4520 T + p^{5} T^{2} \) |
| 83 | \( 1 + 109074 T + p^{5} T^{2} \) |
| 89 | \( 1 - 38490 T + p^{5} T^{2} \) |
| 97 | \( 1 - 1918 T + p^{5} T^{2} \) |
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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
−10.17008314991347024851093134712, −9.638198094674950953685455977866, −8.691623183560660187081764180433, −7.64257352862693891441158175457, −6.81570278441390885381646879589, −5.61092605414331606096215716344, −4.65165374953592395573976834718, −3.04517599026005676930035067922, −2.57799054975783201955172787051, −0.58585769520788832115448019778,
0.58585769520788832115448019778, 2.57799054975783201955172787051, 3.04517599026005676930035067922, 4.65165374953592395573976834718, 5.61092605414331606096215716344, 6.81570278441390885381646879589, 7.64257352862693891441158175457, 8.691623183560660187081764180433, 9.638198094674950953685455977866, 10.17008314991347024851093134712
