L(s) = 1 | + 2-s + 4-s + 2·5-s + 7-s + 8-s + 2·10-s + 6·13-s + 14-s + 16-s + 17-s + 2·20-s + 8·23-s − 25-s + 6·26-s + 28-s − 6·29-s − 8·31-s + 32-s + 34-s + 2·35-s + 10·37-s + 2·40-s − 6·41-s − 12·43-s + 8·46-s + 49-s − 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.447·20-s + 1.66·23-s − 1/5·25-s + 1.17·26-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.171·34-s + 0.338·35-s + 1.64·37-s + 0.316·40-s − 0.937·41-s − 1.82·43-s + 1.17·46-s + 1/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.109117880\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.109117880\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.07773586576360, −12.56892198665102, −11.77578940212500, −11.41238975690061, −11.09673649214852, −10.63289202736973, −10.16831184371354, −9.479213461341216, −9.222422536428181, −8.574362120720928, −8.215712540386928, −7.546885410004172, −7.046182619621615, −6.550694638908190, −6.067889857567735, −5.617356493977830, −5.211754764468737, −4.781562995504043, −3.974557647041201, −3.507598297718075, −3.224096049537216, −2.207483403386293, −1.966435266479437, −1.261375722176592, −0.7047433353092106,
0.7047433353092106, 1.261375722176592, 1.966435266479437, 2.207483403386293, 3.224096049537216, 3.507598297718075, 3.974557647041201, 4.781562995504043, 5.211754764468737, 5.617356493977830, 6.067889857567735, 6.550694638908190, 7.046182619621615, 7.546885410004172, 8.215712540386928, 8.574362120720928, 9.222422536428181, 9.479213461341216, 10.16831184371354, 10.63289202736973, 11.09673649214852, 11.41238975690061, 11.77578940212500, 12.56892198665102, 13.07773586576360