L(s) = 1 | + 3-s + 2·5-s − 4·7-s + 9-s + 11-s − 13-s + 2·15-s + 2·17-s + 4·19-s − 4·21-s − 4·23-s − 25-s + 27-s − 2·29-s + 33-s − 8·35-s − 2·37-s − 39-s + 10·41-s + 4·43-s + 2·45-s + 4·47-s + 9·49-s + 2·51-s + 2·53-s + 2·55-s + 4·57-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 + 0.485·17-s + 0.917·19-s − 0.872·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.174·33-s − 1.35·35-s − 0.328·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.298·45-s + 0.583·47-s + 9/7·49-s + 0.280·51-s + 0.274·53-s + 0.269·55-s + 0.529·57-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
\(L(1)\) |
\(\approx\) |
\(2.506210227\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.506210227\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84295158401243045723072310826, −7.29575157302083845577559018930, −6.50660588022670893045262015766, −5.91295804015695526241953336379, −5.34713009954333227723583604519, −4.12840572278351569621258128511, −3.50765104410569010691848041392, −2.72189393670804729604235133648, −2.00690404101138149837248792729, −0.77673560043025727440487441369,
0.77673560043025727440487441369, 2.00690404101138149837248792729, 2.72189393670804729604235133648, 3.50765104410569010691848041392, 4.12840572278351569621258128511, 5.34713009954333227723583604519, 5.91295804015695526241953336379, 6.50660588022670893045262015766, 7.29575157302083845577559018930, 7.84295158401243045723072310826