| L(s) = 1 | + 5-s + 4·7-s + 2·13-s + 2·17-s − 6·19-s + 6·23-s + 25-s + 6·29-s − 4·31-s + 4·35-s + 4·37-s + 43-s − 10·47-s + 9·49-s + 2·53-s − 4·59-s + 6·61-s + 2·65-s + 12·67-s − 4·73-s − 4·79-s − 6·83-s + 2·85-s + 10·89-s + 8·91-s − 6·95-s + 6·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.554·13-s + 0.485·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.676·35-s + 0.657·37-s + 0.152·43-s − 1.45·47-s + 9/7·49-s + 0.274·53-s − 0.520·59-s + 0.768·61-s + 0.248·65-s + 1.46·67-s − 0.468·73-s − 0.450·79-s − 0.658·83-s + 0.216·85-s + 1.05·89-s + 0.838·91-s − 0.615·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.657153997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.657153997\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.95195852632997, −14.43726764355901, −14.33119804910543, −13.49352849113632, −12.95344651562375, −12.62135188918161, −11.66908057904521, −11.45278349735479, −10.69068667250152, −10.54966872237344, −9.696576852591296, −9.065142234815476, −8.428633371922927, −8.218133216082707, −7.465105464737746, −6.791607532851978, −6.227485601638635, −5.537706863034183, −4.925118683979657, −4.516460658102802, −3.737988396846893, −2.896551853455486, −2.117437978175548, −1.508187188355713, −0.7720648989312279,
0.7720648989312279, 1.508187188355713, 2.117437978175548, 2.896551853455486, 3.737988396846893, 4.516460658102802, 4.925118683979657, 5.537706863034183, 6.227485601638635, 6.791607532851978, 7.465105464737746, 8.218133216082707, 8.428633371922927, 9.065142234815476, 9.696576852591296, 10.54966872237344, 10.69068667250152, 11.45278349735479, 11.66908057904521, 12.62135188918161, 12.95344651562375, 13.49352849113632, 14.33119804910543, 14.43726764355901, 14.95195852632997
