| L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s − 11-s + 15-s + 17-s + 7·19-s + 3·21-s + 6·23-s + 25-s − 27-s − 5·29-s − 2·31-s + 33-s + 3·35-s − 11·37-s + 11·41-s − 2·43-s − 45-s + 5·47-s + 2·49-s − 51-s + 53-s + 55-s − 7·57-s + 10·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.258·15-s + 0.242·17-s + 1.60·19-s + 0.654·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s − 0.359·31-s + 0.174·33-s + 0.507·35-s − 1.80·37-s + 1.71·41-s − 0.304·43-s − 0.149·45-s + 0.729·47-s + 2/7·49-s − 0.140·51-s + 0.137·53-s + 0.134·55-s − 0.927·57-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.816330455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816330455\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.48043366608047, −12.23543847987943, −11.59608453372824, −11.21182089172747, −10.86839179942757, −10.28191444272877, −9.773892516804198, −9.542496653341897, −8.910695947355808, −8.600501194672690, −7.688682338235593, −7.491645935823991, −6.951000746532517, −6.685165348405081, −5.909479533760601, −5.530071673300379, −5.154255442192508, −4.617894373671881, −3.760098726653689, −3.556798223209866, −3.032329149435104, −2.416739790982625, −1.646469821787807, −0.7948364990503612, −0.5092163303555896,
0.5092163303555896, 0.7948364990503612, 1.646469821787807, 2.416739790982625, 3.032329149435104, 3.556798223209866, 3.760098726653689, 4.617894373671881, 5.154255442192508, 5.530071673300379, 5.909479533760601, 6.685165348405081, 6.951000746532517, 7.491645935823991, 7.688682338235593, 8.600501194672690, 8.910695947355808, 9.542496653341897, 9.773892516804198, 10.28191444272877, 10.86839179942757, 11.21182089172747, 11.59608453372824, 12.23543847987943, 12.48043366608047
