L(s) = 1 | − 5-s − 7-s − 3·9-s + 13-s − 2·17-s + 8·19-s − 4·25-s + 9·29-s + 4·31-s + 35-s + 6·37-s − 5·41-s + 8·43-s + 3·45-s + 49-s − 9·53-s − 5·61-s + 3·63-s − 65-s + 16·67-s − 12·71-s − 73-s + 4·79-s + 9·81-s + 12·83-s + 2·85-s + 9·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 0.277·13-s − 0.485·17-s + 1.83·19-s − 4/5·25-s + 1.67·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s − 0.780·41-s + 1.21·43-s + 0.447·45-s + 1/7·49-s − 1.23·53-s − 0.640·61-s + 0.377·63-s − 0.124·65-s + 1.95·67-s − 1.42·71-s − 0.117·73-s + 0.450·79-s + 81-s + 1.31·83-s + 0.216·85-s + 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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
−13.25962294371044, −12.55986423540200, −12.07284533618275, −11.75590854667065, −11.41409279495128, −10.86864027026778, −10.39900897036371, −9.782578897212401, −9.418373491613078, −8.963956358951285, −8.376303316214133, −7.919291592922813, −7.649042886873763, −6.914608278453018, −6.425161107681398, −5.984898261309234, −5.529892978097495, −4.786038667360808, −4.564642270241395, −3.542187371023166, −3.468219889074402, −2.666207724778865, −2.371059850626623, −1.278152193217244, −0.7849369451980740, 0,
0.7849369451980740, 1.278152193217244, 2.371059850626623, 2.666207724778865, 3.468219889074402, 3.542187371023166, 4.564642270241395, 4.786038667360808, 5.529892978097495, 5.984898261309234, 6.425161107681398, 6.914608278453018, 7.649042886873763, 7.919291592922813, 8.376303316214133, 8.963956358951285, 9.418373491613078, 9.782578897212401, 10.39900897036371, 10.86864027026778, 11.41409279495128, 11.75590854667065, 12.07284533618275, 12.55986423540200, 13.25962294371044