L(s) = 1 | − 2·4-s − 2·13-s + 4·16-s + 7·19-s − 5·25-s − 11·31-s − 10·37-s − 13·43-s + 4·52-s + 13·61-s − 8·64-s − 16·67-s + 7·73-s − 14·76-s − 4·79-s − 5·97-s + 10·100-s − 20·103-s − 19·109-s + ⋯ |
L(s) = 1 | − 4-s − 0.554·13-s + 16-s + 1.60·19-s − 25-s − 1.97·31-s − 1.64·37-s − 1.98·43-s + 0.554·52-s + 1.66·61-s − 64-s − 1.95·67-s + 0.819·73-s − 1.60·76-s − 0.450·79-s − 0.507·97-s + 100-s − 1.97·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−9.348577680569783908428726498396, −8.477561354971015169967731719682, −7.66380021222276853297564329904, −6.90688998756717719245467536942, −5.50635879764846365024871064576, −5.18223370651071093866018001979, −3.97618514265534826427174210608, −3.21180480093563962996990541956, −1.62851546714056177560860339476, 0,
1.62851546714056177560860339476, 3.21180480093563962996990541956, 3.97618514265534826427174210608, 5.18223370651071093866018001979, 5.50635879764846365024871064576, 6.90688998756717719245467536942, 7.66380021222276853297564329904, 8.477561354971015169967731719682, 9.348577680569783908428726498396