L(s) = 1 | − 3-s + 4·5-s − 7-s + 9-s + 2·11-s − 6·13-s − 4·15-s − 4·17-s − 4·19-s + 21-s + 2·23-s + 11·25-s − 27-s − 2·29-s − 2·33-s − 4·35-s + 2·37-s + 6·39-s − 4·43-s + 4·45-s + 12·47-s + 49-s + 4·51-s − 6·53-s + 8·55-s + 4·57-s − 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 1.03·15-s − 0.970·17-s − 0.917·19-s + 0.218·21-s + 0.417·23-s + 11/5·25-s − 0.192·27-s − 0.371·29-s − 0.348·33-s − 0.676·35-s + 0.328·37-s + 0.960·39-s − 0.609·43-s + 0.596·45-s + 1.75·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9724619752\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9724619752\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.18271778304860630099542911324, −13.15717312296323701449772388761, −12.33374621805755457062329223302, −10.84044260103233940831848295548, −9.846442211058849071999156153471, −9.094314662852252574100900503933, −6.96207355783957131001717701468, −6.05700480174216171276936327443, −4.80916909916633799382690392302, −2.24239796255929817450421188994,
2.24239796255929817450421188994, 4.80916909916633799382690392302, 6.05700480174216171276936327443, 6.96207355783957131001717701468, 9.094314662852252574100900503933, 9.846442211058849071999156153471, 10.84044260103233940831848295548, 12.33374621805755457062329223302, 13.15717312296323701449772388761, 14.18271778304860630099542911324