L(s) = 1 | − 5-s − 4·7-s − 11-s + 13-s − 2·17-s − 4·19-s + 25-s − 2·29-s − 4·31-s + 4·35-s − 6·37-s − 2·41-s − 4·43-s + 9·49-s + 6·53-s + 55-s − 4·59-s − 2·61-s − 65-s + 12·67-s + 10·73-s + 4·77-s + 8·79-s − 12·83-s + 2·85-s + 6·89-s − 4·91-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.301·11-s + 0.277·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 9/7·49-s + 0.824·53-s + 0.134·55-s − 0.520·59-s − 0.256·61-s − 0.124·65-s + 1.46·67-s + 1.17·73-s + 0.455·77-s + 0.900·79-s − 1.31·83-s + 0.216·85-s + 0.635·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−14.09693545163895, −13.59653583018554, −13.20912761701669, −12.73785666813590, −12.36669623219525, −11.90695771164521, −11.15052154135898, −10.79432412920144, −10.34532692467666, −9.709735685092446, −9.346346254024201, −8.747454793748052, −8.333300447513373, −7.728959923528519, −7.027388517091630, −6.661872038442897, −6.317287671699815, −5.512777481533317, −5.142271065129858, −4.236875031310579, −3.795849618303774, −3.361337446071899, −2.651510714765202, −2.099477065200590, −1.177988436113182, 0, 0,
1.177988436113182, 2.099477065200590, 2.651510714765202, 3.361337446071899, 3.795849618303774, 4.236875031310579, 5.142271065129858, 5.512777481533317, 6.317287671699815, 6.661872038442897, 7.027388517091630, 7.728959923528519, 8.333300447513373, 8.747454793748052, 9.346346254024201, 9.709735685092446, 10.34532692467666, 10.79432412920144, 11.15052154135898, 11.90695771164521, 12.36669623219525, 12.73785666813590, 13.20912761701669, 13.59653583018554, 14.09693545163895