| L(s) = 1 | − 2·7-s − 14·13-s − 14·19-s + 22·31-s − 20·37-s − 26·43-s − 11·49-s − 2·61-s + 22·67-s − 20·73-s − 8·79-s + 28·91-s − 38·97-s + 40·103-s + 34·109-s − 22·121-s + 127-s + 131-s + 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 121·169-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 3.88·13-s − 3.21·19-s + 3.95·31-s − 3.28·37-s − 3.96·43-s − 1.57·49-s − 0.256·61-s + 2.68·67-s − 2.34·73-s − 0.900·79-s + 2.93·91-s − 3.85·97-s + 3.94·103-s + 3.25·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 2.42·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.71840331908472710868523661717, −12.19470428841518974341961809081, −12.19470428841518974341961809081, −11.39681998035064824319938848751, −11.39681998035064824319938848751, −10.14453650340834346363527657164, −10.14453650340834346363527657164, −10.01185297753129750762994211064, −10.01185297753129750762994211064, −8.810318287138666656620561570522, −8.810318287138666656620561570522, −8.068267125649044096169699508794, −8.068267125649044096169699508794, −7.00418644170583896469357710759, −7.00418644170583896469357710759, −6.41391381288026324076920371033, −6.41391381288026324076920371033, −5.13958532772580839484877653078, −5.13958532772580839484877653078, −4.42116330171299264580891094162, −4.42116330171299264580891094162, −3.08601112889137016020769949109, −3.08601112889137016020769949109, −2.05095977025298622950154668732, −2.05095977025298622950154668732, 0, 0,
2.05095977025298622950154668732, 2.05095977025298622950154668732, 3.08601112889137016020769949109, 3.08601112889137016020769949109, 4.42116330171299264580891094162, 4.42116330171299264580891094162, 5.13958532772580839484877653078, 5.13958532772580839484877653078, 6.41391381288026324076920371033, 6.41391381288026324076920371033, 7.00418644170583896469357710759, 7.00418644170583896469357710759, 8.068267125649044096169699508794, 8.068267125649044096169699508794, 8.810318287138666656620561570522, 8.810318287138666656620561570522, 10.01185297753129750762994211064, 10.01185297753129750762994211064, 10.14453650340834346363527657164, 10.14453650340834346363527657164, 11.39681998035064824319938848751, 11.39681998035064824319938848751, 12.19470428841518974341961809081, 12.19470428841518974341961809081, 12.71840331908472710868523661717
