| L(s) = 1 | − 5-s + 2·7-s − 4·11-s + 4·17-s + 6·19-s + 25-s + 4·29-s + 6·31-s − 2·35-s + 2·37-s + 10·41-s − 8·43-s − 3·49-s + 4·53-s + 4·55-s − 4·59-s + 2·61-s + 6·67-s + 8·71-s − 10·73-s − 8·77-s + 4·79-s + 12·83-s − 4·85-s − 2·89-s − 6·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 1.20·11-s + 0.970·17-s + 1.37·19-s + 1/5·25-s + 0.742·29-s + 1.07·31-s − 0.338·35-s + 0.328·37-s + 1.56·41-s − 1.21·43-s − 3/7·49-s + 0.549·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.733·67-s + 0.949·71-s − 1.17·73-s − 0.911·77-s + 0.450·79-s + 1.31·83-s − 0.433·85-s − 0.211·89-s − 0.615·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.956041964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.956041964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.57365377188804, −13.14027719506526, −12.42721431297094, −12.14160076085118, −11.62384980445826, −11.15773716661699, −10.72571490776468, −10.06208025042174, −9.821573332311572, −9.184454252968022, −8.385443713612966, −8.106086815619284, −7.720710168280708, −7.269359281503079, −6.638309609286651, −5.871723910667447, −5.462088744141479, −4.842242031098704, −4.599068501217231, −3.727229699713981, −3.121709619103650, −2.705824835871637, −1.935131648453142, −1.080005500294733, −0.6093685390143791,
0.6093685390143791, 1.080005500294733, 1.935131648453142, 2.705824835871637, 3.121709619103650, 3.727229699713981, 4.599068501217231, 4.842242031098704, 5.462088744141479, 5.871723910667447, 6.638309609286651, 7.269359281503079, 7.720710168280708, 8.106086815619284, 8.385443713612966, 9.184454252968022, 9.821573332311572, 10.06208025042174, 10.72571490776468, 11.15773716661699, 11.62384980445826, 12.14160076085118, 12.42721431297094, 13.14027719506526, 13.57365377188804
