| L(s) = 1 | + 7·5-s + 26·7-s − 11·11-s + 52·13-s − 46·17-s + 96·19-s + 27·23-s − 76·25-s − 16·29-s + 293·31-s + 182·35-s − 29·37-s + 472·41-s + 110·43-s − 224·47-s + 333·49-s − 754·53-s − 77·55-s + 825·59-s − 548·61-s + 364·65-s + 123·67-s + 1.00e3·71-s − 1.02e3·73-s − 286·77-s − 526·79-s − 158·83-s + ⋯ |
| L(s) = 1 | + 0.626·5-s + 1.40·7-s − 0.301·11-s + 1.10·13-s − 0.656·17-s + 1.15·19-s + 0.244·23-s − 0.607·25-s − 0.102·29-s + 1.69·31-s + 0.878·35-s − 0.128·37-s + 1.79·41-s + 0.390·43-s − 0.695·47-s + 0.970·49-s − 1.95·53-s − 0.188·55-s + 1.82·59-s − 1.15·61-s + 0.694·65-s + 0.224·67-s + 1.67·71-s − 1.63·73-s − 0.423·77-s − 0.749·79-s − 0.208·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.379481007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.379481007\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 96 T + p^{3} T^{2} \) |
| 23 | \( 1 - 27 T + p^{3} T^{2} \) |
| 29 | \( 1 + 16 T + p^{3} T^{2} \) |
| 31 | \( 1 - 293 T + p^{3} T^{2} \) |
| 37 | \( 1 + 29 T + p^{3} T^{2} \) |
| 41 | \( 1 - 472 T + p^{3} T^{2} \) |
| 43 | \( 1 - 110 T + p^{3} T^{2} \) |
| 47 | \( 1 + 224 T + p^{3} T^{2} \) |
| 53 | \( 1 + 754 T + p^{3} T^{2} \) |
| 59 | \( 1 - 825 T + p^{3} T^{2} \) |
| 61 | \( 1 + 548 T + p^{3} T^{2} \) |
| 67 | \( 1 - 123 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1001 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 79 | \( 1 + 526 T + p^{3} T^{2} \) |
| 83 | \( 1 + 158 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1217 T + p^{3} T^{2} \) |
| 97 | \( 1 + 263 T + p^{3} T^{2} \) |
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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
−9.036972369585110320218302208486, −8.176264238368301283270321901260, −7.68056424196734473240759287247, −6.52538982442185044380895069746, −5.74674643230168395855459288922, −4.95420751869720998209705587932, −4.14134783362483100766259245313, −2.86781227021253339178570339667, −1.80472589896682391374798968624, −0.961190492282931116715747112878,
0.961190492282931116715747112878, 1.80472589896682391374798968624, 2.86781227021253339178570339667, 4.14134783362483100766259245313, 4.95420751869720998209705587932, 5.74674643230168395855459288922, 6.52538982442185044380895069746, 7.68056424196734473240759287247, 8.176264238368301283270321901260, 9.036972369585110320218302208486
