| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 4·11-s + 6·13-s + 15-s − 4·17-s − 21-s − 23-s + 25-s − 27-s + 4·33-s − 35-s − 6·37-s − 6·39-s − 45-s − 8·47-s + 49-s + 4·51-s + 8·53-s + 4·55-s + 10·61-s + 63-s − 6·65-s + 8·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s − 0.970·17-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.696·33-s − 0.169·35-s − 0.986·37-s − 0.960·39-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.560·51-s + 1.09·53-s + 0.539·55-s + 1.28·61-s + 0.125·63-s − 0.744·65-s + 0.977·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.25707617633591, −14.67016601800357, −13.84578819254819, −13.54465644845057, −12.89591966873907, −12.65468233053367, −11.71049346234241, −11.43597597998910, −10.93419223679157, −10.49054990717020, −10.01523116555286, −9.148091674686286, −8.510236127972320, −8.246265823779667, −7.589530016879951, −6.861890593956237, −6.463911255499914, −5.681062512756339, −5.253983776330580, −4.607867341146296, −3.934827699482452, −3.411049469442680, −2.486988355023541, −1.762831742078469, −0.8795912219916253, 0,
0.8795912219916253, 1.762831742078469, 2.486988355023541, 3.411049469442680, 3.934827699482452, 4.607867341146296, 5.253983776330580, 5.681062512756339, 6.463911255499914, 6.861890593956237, 7.589530016879951, 8.246265823779667, 8.510236127972320, 9.148091674686286, 10.01523116555286, 10.49054990717020, 10.93419223679157, 11.43597597998910, 11.71049346234241, 12.65468233053367, 12.89591966873907, 13.54465644845057, 13.84578819254819, 14.67016601800357, 15.25707617633591
