| L(s) = 1 | − 3·5-s − 7-s + 3·11-s + 17-s + 19-s + 7·23-s + 4·25-s − 5·29-s − 6·31-s + 3·35-s + 3·37-s + 2·41-s + 43-s + 49-s + 2·53-s − 9·55-s − 10·59-s + 11·61-s − 8·67-s − 2·71-s + 7·73-s − 3·77-s + 10·79-s − 14·83-s − 3·85-s + 2·89-s − 3·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 0.904·11-s + 0.242·17-s + 0.229·19-s + 1.45·23-s + 4/5·25-s − 0.928·29-s − 1.07·31-s + 0.507·35-s + 0.493·37-s + 0.312·41-s + 0.152·43-s + 1/7·49-s + 0.274·53-s − 1.21·55-s − 1.30·59-s + 1.40·61-s − 0.977·67-s − 0.237·71-s + 0.819·73-s − 0.341·77-s + 1.12·79-s − 1.53·83-s − 0.325·85-s + 0.211·89-s − 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.498450006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498450006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.04993864502653, −13.23176112412737, −12.96639986029658, −12.35715357572403, −11.95054067817558, −11.46706464473486, −10.98708616299922, −10.71617405811779, −9.762894819011206, −9.406946539851201, −8.887442936712613, −8.432190963790413, −7.699576030530697, −7.345863376232229, −6.940223432103186, −6.291914171455545, −5.651988799361420, −5.049306415461490, −4.358707074672363, −3.892752007935460, −3.392027400611725, −2.911721476968925, −1.955622301824019, −1.127946857035426, −0.4481808569316392,
0.4481808569316392, 1.127946857035426, 1.955622301824019, 2.911721476968925, 3.392027400611725, 3.892752007935460, 4.358707074672363, 5.049306415461490, 5.651988799361420, 6.291914171455545, 6.940223432103186, 7.345863376232229, 7.699576030530697, 8.432190963790413, 8.887442936712613, 9.406946539851201, 9.762894819011206, 10.71617405811779, 10.98708616299922, 11.46706464473486, 11.95054067817558, 12.35715357572403, 12.96639986029658, 13.23176112412737, 14.04993864502653
