L(s) = 1 | − 3-s + 9-s − 6·11-s + 4·13-s − 3·17-s − 4·19-s − 3·23-s − 27-s + 6·29-s − 5·31-s + 6·33-s + 8·37-s − 4·39-s + 3·41-s + 8·43-s + 9·47-s + 3·51-s + 12·53-s + 4·57-s + 6·59-s + 2·61-s + 8·67-s + 3·69-s − 9·71-s + 14·73-s − 7·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 0.727·17-s − 0.917·19-s − 0.625·23-s − 0.192·27-s + 1.11·29-s − 0.898·31-s + 1.04·33-s + 1.31·37-s − 0.640·39-s + 0.468·41-s + 1.21·43-s + 1.31·47-s + 0.420·51-s + 1.64·53-s + 0.529·57-s + 0.781·59-s + 0.256·61-s + 0.977·67-s + 0.361·69-s − 1.06·71-s + 1.63·73-s − 0.787·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.934385976\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934385976\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−12.83257387794974, −12.66573920851430, −11.94014146524046, −11.45065169259467, −10.87404297960489, −10.73492445710931, −10.28964395637374, −9.794058890268623, −9.107361708664390, −8.588358863870650, −8.294953162137717, −7.629249638143428, −7.317606904154528, −6.617280196174534, −6.108989198607146, −5.765182322077424, −5.278679299701699, −4.668797815388282, −4.127506620447802, −3.781487649906628, −2.838620179752900, −2.374292012560700, −1.958902383389192, −0.8503462315790333, −0.5142103144341542,
0.5142103144341542, 0.8503462315790333, 1.958902383389192, 2.374292012560700, 2.838620179752900, 3.781487649906628, 4.127506620447802, 4.668797815388282, 5.278679299701699, 5.765182322077424, 6.108989198607146, 6.617280196174534, 7.317606904154528, 7.629249638143428, 8.294953162137717, 8.588358863870650, 9.107361708664390, 9.794058890268623, 10.28964395637374, 10.73492445710931, 10.87404297960489, 11.45065169259467, 11.94014146524046, 12.66573920851430, 12.83257387794974