| L(s) = 1 | + 2-s − 4-s + 5-s + 2·7-s − 3·8-s + 10-s − 13-s + 2·14-s − 16-s + 2·19-s − 20-s + 25-s − 26-s − 2·28-s − 10·29-s + 4·31-s + 5·32-s + 2·35-s − 2·37-s + 2·38-s − 3·40-s + 6·41-s + 6·43-s + 8·47-s − 3·49-s + 50-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s + 0.755·7-s − 1.06·8-s + 0.316·10-s − 0.277·13-s + 0.534·14-s − 1/4·16-s + 0.458·19-s − 0.223·20-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 1.85·29-s + 0.718·31-s + 0.883·32-s + 0.338·35-s − 0.328·37-s + 0.324·38-s − 0.474·40-s + 0.937·41-s + 0.914·43-s + 1.16·47-s − 3/7·49-s + 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.51937833238189, −13.05774455360155, −12.53601759299268, −12.22329783245640, −11.58571636641631, −11.27750167540237, −10.55813254215541, −10.25997523815464, −9.474912639016067, −9.093557266628791, −8.958157733220618, −8.036104770973135, −7.689058578487162, −7.248340430168602, −6.425239358647370, −5.916780881257067, −5.576922000046205, −5.044167281302004, −4.498724630621029, −4.161220627364673, −3.431600750513946, −2.910336131277579, −2.257584415210602, −1.600525260006126, −0.8744113330694292, 0,
0.8744113330694292, 1.600525260006126, 2.257584415210602, 2.910336131277579, 3.431600750513946, 4.161220627364673, 4.498724630621029, 5.044167281302004, 5.576922000046205, 5.916780881257067, 6.425239358647370, 7.248340430168602, 7.689058578487162, 8.036104770973135, 8.958157733220618, 9.093557266628791, 9.474912639016067, 10.25997523815464, 10.55813254215541, 11.27750167540237, 11.58571636641631, 12.22329783245640, 12.53601759299268, 13.05774455360155, 13.51937833238189
