| L(s) = 1 | − 3-s − 5·7-s + 9-s − 11-s − 13-s − 17-s − 7·19-s + 5·21-s − 3·23-s − 27-s + 6·29-s − 31-s + 33-s + 7·37-s + 39-s − 12·41-s − 12·43-s − 4·47-s + 18·49-s + 51-s − 10·53-s + 7·57-s − 4·59-s + 11·61-s − 5·63-s − 3·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.242·17-s − 1.60·19-s + 1.09·21-s − 0.625·23-s − 0.192·27-s + 1.11·29-s − 0.179·31-s + 0.174·33-s + 1.15·37-s + 0.160·39-s − 1.87·41-s − 1.82·43-s − 0.583·47-s + 18/7·49-s + 0.140·51-s − 1.37·53-s + 0.927·57-s − 0.520·59-s + 1.40·61-s − 0.629·63-s − 0.366·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05185912109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05185912109\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| 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
−12.84783442052594, −12.56760302359063, −12.15320554615065, −11.56573742364046, −11.13928399043120, −10.40498237646136, −10.19379393003823, −9.823621152517491, −9.372496385236814, −8.659802724843212, −8.344098260119210, −7.722038582002387, −6.995676245872093, −6.606432220965249, −6.267779112219325, −6.041386688245183, −5.073275316331471, −4.828821525607229, −4.112901412243363, −3.493476357667015, −3.177382326195138, −2.347493627663806, −1.971316891739616, −0.9271263736823885, −0.07648309359481235,
0.07648309359481235, 0.9271263736823885, 1.971316891739616, 2.347493627663806, 3.177382326195138, 3.493476357667015, 4.112901412243363, 4.828821525607229, 5.073275316331471, 6.041386688245183, 6.267779112219325, 6.606432220965249, 6.995676245872093, 7.722038582002387, 8.344098260119210, 8.659802724843212, 9.372496385236814, 9.823621152517491, 10.19379393003823, 10.40498237646136, 11.13928399043120, 11.56573742364046, 12.15320554615065, 12.56760302359063, 12.84783442052594
