L(s) = 1 | − 3-s + 7-s + 9-s + 2·11-s − 6·13-s + 2·17-s + 6·19-s − 21-s − 23-s − 5·25-s − 27-s − 6·29-s − 2·33-s + 6·39-s + 6·41-s + 6·43-s − 8·47-s + 49-s − 2·51-s − 4·53-s − 6·57-s − 8·61-s + 63-s + 2·67-s + 69-s − 2·73-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 0.485·17-s + 1.37·19-s − 0.218·21-s − 0.208·23-s − 25-s − 0.192·27-s − 1.11·29-s − 0.348·33-s + 0.960·39-s + 0.937·41-s + 0.914·43-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.549·53-s − 0.794·57-s − 1.02·61-s + 0.125·63-s + 0.244·67-s + 0.120·69-s − 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.60839998497940622646495106389, −6.92319938252210548801341388292, −5.98308017575993809356305078263, −5.44418806943124157061745404643, −4.77641761339239800879487035357, −4.04846806507089455955874915378, −3.13926693599211134068390463128, −2.13714269448030660340735657450, −1.22951817597761996846268232936, 0,
1.22951817597761996846268232936, 2.13714269448030660340735657450, 3.13926693599211134068390463128, 4.04846806507089455955874915378, 4.77641761339239800879487035357, 5.44418806943124157061745404643, 5.98308017575993809356305078263, 6.92319938252210548801341388292, 7.60839998497940622646495106389