| L(s) = 1 | − 3-s − 2·5-s + 9-s − 2·11-s + 4·13-s + 2·15-s + 6·17-s − 8·19-s − 6·23-s − 25-s − 27-s + 10·29-s + 4·31-s + 2·33-s − 6·37-s − 4·39-s − 6·41-s − 4·43-s − 2·45-s + 8·47-s − 6·51-s − 2·53-s + 4·55-s + 8·57-s + 4·59-s + 8·61-s − 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s + 1.10·13-s + 0.516·15-s + 1.45·17-s − 1.83·19-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.348·33-s − 0.986·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s − 0.840·51-s − 0.274·53-s + 0.539·55-s + 1.05·57-s + 0.520·59-s + 1.02·61-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9941235985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9941235985\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.84733514344217811543637145675, −6.92463517275062224467790787245, −6.33416786127188162365616722367, −5.71413978678226731439575001600, −4.93576851597948606357679700125, −4.11040677188913589025107761424, −3.67628326977236520379423098567, −2.67326703904525241086609724462, −1.57152773940275667407432020188, −0.50379999108043513770100986886,
0.50379999108043513770100986886, 1.57152773940275667407432020188, 2.67326703904525241086609724462, 3.67628326977236520379423098567, 4.11040677188913589025107761424, 4.93576851597948606357679700125, 5.71413978678226731439575001600, 6.33416786127188162365616722367, 6.92463517275062224467790787245, 7.84733514344217811543637145675
