L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 11-s + 13-s − 2·15-s − 6·17-s + 4·21-s − 8·23-s − 25-s − 27-s − 2·29-s − 4·31-s + 33-s − 8·35-s + 6·37-s − 39-s − 2·41-s + 4·43-s + 2·45-s + 9·49-s + 6·51-s + 6·53-s − 2·55-s + 12·59-s + 14·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.516·15-s − 1.45·17-s + 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.174·33-s − 1.35·35-s + 0.986·37-s − 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 9/7·49-s + 0.840·51-s + 0.824·53-s − 0.269·55-s + 1.56·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9252306911\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9252306911\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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
−7.914185518947526550070135441399, −6.94920643056734156582492173258, −6.51214529910564456330900838635, −5.85531253282042441977212173551, −5.46860945052884866910076873817, −4.26195892061379619291597682602, −3.71676304637866275961768286271, −2.54962755576930928992918774887, −1.95787456420318355282352858787, −0.47853428559794312090917054749,
0.47853428559794312090917054749, 1.95787456420318355282352858787, 2.54962755576930928992918774887, 3.71676304637866275961768286271, 4.26195892061379619291597682602, 5.46860945052884866910076873817, 5.85531253282042441977212173551, 6.51214529910564456330900838635, 6.94920643056734156582492173258, 7.914185518947526550070135441399