L(s) = 1 | − 4·5-s + 3·7-s + 4·11-s + 13-s + 4·17-s + 19-s + 4·23-s + 11·25-s + 4·31-s − 12·35-s − 9·37-s + 8·43-s − 12·47-s + 2·49-s + 8·53-s − 16·55-s + 4·59-s − 5·61-s − 4·65-s − 11·67-s + 8·71-s + 73-s + 12·77-s + 5·79-s + 8·83-s − 16·85-s − 12·89-s + ⋯ |
L(s) = 1 | − 1.78·5-s + 1.13·7-s + 1.20·11-s + 0.277·13-s + 0.970·17-s + 0.229·19-s + 0.834·23-s + 11/5·25-s + 0.718·31-s − 2.02·35-s − 1.47·37-s + 1.21·43-s − 1.75·47-s + 2/7·49-s + 1.09·53-s − 2.15·55-s + 0.520·59-s − 0.640·61-s − 0.496·65-s − 1.34·67-s + 0.949·71-s + 0.117·73-s + 1.36·77-s + 0.562·79-s + 0.878·83-s − 1.73·85-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.261655679\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.261655679\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−11.41833349739553039502799551308, −10.52317198885866389759410232053, −9.093326194542073314647733546631, −8.285403607109998915279053043958, −7.61689482228550235915151273028, −6.71788542885177681871412514582, −5.14859535448368711259372949904, −4.20575968417257290967879965799, −3.33304684468240638767778076253, −1.18156952361835473635888524724,
1.18156952361835473635888524724, 3.33304684468240638767778076253, 4.20575968417257290967879965799, 5.14859535448368711259372949904, 6.71788542885177681871412514582, 7.61689482228550235915151273028, 8.285403607109998915279053043958, 9.093326194542073314647733546631, 10.52317198885866389759410232053, 11.41833349739553039502799551308