L(s) = 1 | + 3-s − 2·5-s + 9-s + 4·11-s − 2·13-s − 2·15-s − 17-s + 4·19-s − 25-s + 27-s + 10·29-s − 8·31-s + 4·33-s + 2·37-s − 2·39-s − 10·41-s − 12·43-s − 2·45-s − 51-s − 6·53-s − 8·55-s + 4·57-s + 12·59-s − 10·61-s + 4·65-s + 12·67-s − 10·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.320·39-s − 1.56·41-s − 1.82·43-s − 0.298·45-s − 0.140·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 1.56·59-s − 1.28·61-s + 0.496·65-s + 1.46·67-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.54672689325630, −13.09179148555876, −12.48044408050806, −11.99135271880035, −11.66420880266146, −11.39696779179956, −10.62431322042630, −10.00187421142739, −9.772079485155243, −9.109889051541514, −8.656800581549635, −8.277300400451674, −7.702789927065238, −7.245137953029305, −6.746684734057688, −6.393990573287284, −5.548059242155008, −4.918196818202853, −4.510755927228541, −3.886316755749253, −3.353430447496017, −3.084922600176579, −2.124228643432955, −1.596715652747368, −0.8523915268642746, 0,
0.8523915268642746, 1.596715652747368, 2.124228643432955, 3.084922600176579, 3.353430447496017, 3.886316755749253, 4.510755927228541, 4.918196818202853, 5.548059242155008, 6.393990573287284, 6.746684734057688, 7.245137953029305, 7.702789927065238, 8.277300400451674, 8.656800581549635, 9.109889051541514, 9.772079485155243, 10.00187421142739, 10.62431322042630, 11.39696779179956, 11.66420880266146, 11.99135271880035, 12.48044408050806, 13.09179148555876, 13.54672689325630