| L(s) = 1 | + 3-s + 9-s + 6·11-s + 13-s + 3·17-s + 4·19-s − 3·23-s + 27-s − 3·29-s + 5·31-s + 6·33-s − 10·37-s + 39-s − 9·41-s + 43-s + 3·51-s + 9·53-s + 4·57-s − 9·59-s + 11·61-s + 4·67-s − 3·69-s + 12·71-s − 10·73-s + 10·79-s + 81-s + 9·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s − 0.625·23-s + 0.192·27-s − 0.557·29-s + 0.898·31-s + 1.04·33-s − 1.64·37-s + 0.160·39-s − 1.40·41-s + 0.152·43-s + 0.420·51-s + 1.23·53-s + 0.529·57-s − 1.17·59-s + 1.40·61-s + 0.488·67-s − 0.361·69-s + 1.42·71-s − 1.17·73-s + 1.12·79-s + 1/9·81-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.728758074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.728758074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.98738268597614, −12.13976595706973, −12.04651096853850, −11.78918054046938, −11.04564728224984, −10.55214724700482, −9.933503491268963, −9.680244473049497, −9.113887032845461, −8.788072928277198, −8.186607868619128, −7.846361442639830, −7.158177042290098, −6.685253908303322, −6.457233691528741, −5.588823666654472, −5.299584161037886, −4.556320813070930, −3.899293815367184, −3.608125820653958, −3.200589822688121, −2.370743536443559, −1.703338759743397, −1.281127680855632, −0.6029894504941901,
0.6029894504941901, 1.281127680855632, 1.703338759743397, 2.370743536443559, 3.200589822688121, 3.608125820653958, 3.899293815367184, 4.556320813070930, 5.299584161037886, 5.588823666654472, 6.457233691528741, 6.685253908303322, 7.158177042290098, 7.846361442639830, 8.186607868619128, 8.788072928277198, 9.113887032845461, 9.680244473049497, 9.933503491268963, 10.55214724700482, 11.04564728224984, 11.78918054046938, 12.04651096853850, 12.13976595706973, 12.98738268597614
