| L(s) = 1 | − 2-s − 7·4-s − 5·5-s − 6·7-s + 15·8-s + 5·10-s + 47·11-s − 5·13-s + 6·14-s + 41·16-s + 131·17-s − 56·19-s + 35·20-s − 47·22-s − 3·23-s + 25·25-s + 5·26-s + 42·28-s + 157·29-s + 225·31-s − 161·32-s − 131·34-s + 30·35-s − 70·37-s + 56·38-s − 75·40-s − 140·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s − 0.447·5-s − 0.323·7-s + 0.662·8-s + 0.158·10-s + 1.28·11-s − 0.106·13-s + 0.114·14-s + 0.640·16-s + 1.86·17-s − 0.676·19-s + 0.391·20-s − 0.455·22-s − 0.0271·23-s + 1/5·25-s + 0.0377·26-s + 0.283·28-s + 1.00·29-s + 1.30·31-s − 0.889·32-s − 0.660·34-s + 0.144·35-s − 0.311·37-s + 0.239·38-s − 0.296·40-s − 0.533·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.048689778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048689778\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 47 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 T + p^{3} T^{2} \) |
| 17 | \( 1 - 131 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + 3 T + p^{3} T^{2} \) |
| 29 | \( 1 - 157 T + p^{3} T^{2} \) |
| 31 | \( 1 - 225 T + p^{3} T^{2} \) |
| 37 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 + 140 T + p^{3} T^{2} \) |
| 43 | \( 1 - 397 T + p^{3} T^{2} \) |
| 47 | \( 1 - 347 T + p^{3} T^{2} \) |
| 53 | \( 1 + 4 T + p^{3} T^{2} \) |
| 59 | \( 1 + 748 T + p^{3} T^{2} \) |
| 61 | \( 1 + 338 T + p^{3} T^{2} \) |
| 67 | \( 1 - 492 T + p^{3} T^{2} \) |
| 71 | \( 1 + 32 T + p^{3} T^{2} \) |
| 73 | \( 1 - 970 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1257 T + p^{3} T^{2} \) |
| 83 | \( 1 - 102 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1488 T + p^{3} T^{2} \) |
| 97 | \( 1 - 974 T + p^{3} T^{2} \) |
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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
−12.58562658321373369174771965047, −11.93248616048285875174257002557, −10.45916150004869241692768592164, −9.582825343835845548955191130502, −8.611716074921356847310242411565, −7.61012881132716336454627823555, −6.18924663374624560601395957921, −4.64482899482612480145519690979, −3.49117096421590371752233322471, −0.961465066411171040216434074886,
0.961465066411171040216434074886, 3.49117096421590371752233322471, 4.64482899482612480145519690979, 6.18924663374624560601395957921, 7.61012881132716336454627823555, 8.611716074921356847310242411565, 9.582825343835845548955191130502, 10.45916150004869241692768592164, 11.93248616048285875174257002557, 12.58562658321373369174771965047
