| L(s) = 1 | − 2·5-s − 7-s + 2·11-s − 13-s + 6·17-s − 5·19-s + 6·23-s − 25-s − 8·29-s − 8·31-s + 2·35-s + 5·37-s + 8·41-s − 4·43-s − 10·47-s − 6·49-s − 4·53-s − 4·55-s − 14·59-s − 3·61-s + 2·65-s − 13·67-s − 4·71-s + 9·73-s − 2·77-s − 11·79-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.603·11-s − 0.277·13-s + 1.45·17-s − 1.14·19-s + 1.25·23-s − 1/5·25-s − 1.48·29-s − 1.43·31-s + 0.338·35-s + 0.821·37-s + 1.24·41-s − 0.609·43-s − 1.45·47-s − 6/7·49-s − 0.549·53-s − 0.539·55-s − 1.82·59-s − 0.384·61-s + 0.248·65-s − 1.58·67-s − 0.474·71-s + 1.05·73-s − 0.227·77-s − 1.23·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−9.091038044170517247847576338371, −7.908747723265166312221216125440, −7.53518840978743106177330231964, −6.56129874280569123918857312585, −5.72569558138507357075854697581, −4.68301032913957237602385420184, −3.77298647544947819109929811146, −3.09100156780521462109801579892, −1.58075024630036371346424385734, 0,
1.58075024630036371346424385734, 3.09100156780521462109801579892, 3.77298647544947819109929811146, 4.68301032913957237602385420184, 5.72569558138507357075854697581, 6.56129874280569123918857312585, 7.53518840978743106177330231964, 7.908747723265166312221216125440, 9.091038044170517247847576338371
