| L(s) = 1 | − 2-s + 4-s + 5-s + 4·7-s − 8-s − 10-s − 3·11-s + 13-s − 4·14-s + 16-s − 19-s + 20-s + 3·22-s − 6·23-s + 25-s − 26-s + 4·28-s − 5·31-s − 32-s + 4·35-s − 11·37-s + 38-s − 40-s − 6·41-s + 5·43-s − 3·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.229·19-s + 0.223·20-s + 0.639·22-s − 1.25·23-s + 1/5·25-s − 0.196·26-s + 0.755·28-s − 0.898·31-s − 0.176·32-s + 0.676·35-s − 1.80·37-s + 0.162·38-s − 0.158·40-s − 0.937·41-s + 0.762·43-s − 0.452·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.982585202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.982585202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.42289777071501, −11.92084302039416, −11.77089754923329, −11.00624021982411, −10.66469235257899, −10.42385518264083, −9.976393443153590, −9.315512986055031, −8.773297002621061, −8.477188091794708, −8.130684385211111, −7.449388346242552, −7.292586368545631, −6.662257870310743, −5.893251910466304, −5.586314710236665, −5.169213005457501, −4.627694487870399, −3.910223607224312, −3.533889550051952, −2.555541466498022, −2.161843661615003, −1.816565143555713, −1.131787134052676, −0.4242499595439938,
0.4242499595439938, 1.131787134052676, 1.816565143555713, 2.161843661615003, 2.555541466498022, 3.533889550051952, 3.910223607224312, 4.627694487870399, 5.169213005457501, 5.586314710236665, 5.893251910466304, 6.662257870310743, 7.292586368545631, 7.449388346242552, 8.130684385211111, 8.477188091794708, 8.773297002621061, 9.315512986055031, 9.976393443153590, 10.42385518264083, 10.66469235257899, 11.00624021982411, 11.77089754923329, 11.92084302039416, 12.42289777071501
