| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 13-s − 14-s + 16-s + 3·17-s − 20-s − 3·23-s + 25-s + 26-s − 28-s − 3·29-s − 2·31-s + 32-s + 3·34-s + 35-s + 10·37-s − 40-s + 6·41-s + 2·43-s − 3·46-s − 6·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.223·20-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 0.557·29-s − 0.359·31-s + 0.176·32-s + 0.514·34-s + 0.169·35-s + 1.64·37-s − 0.158·40-s + 0.937·41-s + 0.304·43-s − 0.442·46-s − 6/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.135108757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.135108757\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.87034499579530, −14.50197713581746, −14.14612311486042, −13.25111233496736, −13.02525179250370, −12.48845416692204, −11.90300892328573, −11.36716736819082, −11.00932154311092, −10.23926943029560, −9.748173466182201, −9.194370968534208, −8.384427846133160, −7.870906244919801, −7.371816453856856, −6.722913090790210, −6.077000972574461, −5.620815198698268, −4.971956646276204, −4.059992988476425, −3.897224677989712, −3.021840741817264, −2.465375192272983, −1.513140317058226, −0.6049779824549590,
0.6049779824549590, 1.513140317058226, 2.465375192272983, 3.021840741817264, 3.897224677989712, 4.059992988476425, 4.971956646276204, 5.620815198698268, 6.077000972574461, 6.722913090790210, 7.371816453856856, 7.870906244919801, 8.384427846133160, 9.194370968534208, 9.748173466182201, 10.23926943029560, 11.00932154311092, 11.36716736819082, 11.90300892328573, 12.48845416692204, 13.02525179250370, 13.25111233496736, 14.14612311486042, 14.50197713581746, 14.87034499579530
