| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 4·11-s + 13-s − 14-s + 16-s + 3·17-s − 19-s + 4·22-s + 26-s − 28-s − 29-s + 10·31-s + 32-s + 3·34-s + 2·37-s − 38-s + 4·41-s − 6·43-s + 4·44-s − 6·49-s + 52-s − 53-s − 56-s − 58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.20·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.852·22-s + 0.196·26-s − 0.188·28-s − 0.185·29-s + 1.79·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s − 0.162·38-s + 0.624·41-s − 0.914·43-s + 0.603·44-s − 6/7·49-s + 0.138·52-s − 0.137·53-s − 0.133·56-s − 0.131·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.520649333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.520649333\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.23152491757729, −12.78473328130168, −12.18706923809562, −11.98547771694963, −11.34854823164315, −11.10979302513560, −10.30688674980187, −9.941326570560678, −9.507037814950760, −8.933205475999166, −8.321950569478621, −7.950163239439730, −7.277256315093645, −6.698977095541945, −6.348157066159785, −5.991905802775225, −5.281641628379971, −4.756638119362646, −4.205534737616093, −3.676963993128287, −3.248504049764479, −2.620076680327035, −1.919955127161359, −1.221711714282925, −0.6442434122838454,
0.6442434122838454, 1.221711714282925, 1.919955127161359, 2.620076680327035, 3.248504049764479, 3.676963993128287, 4.205534737616093, 4.756638119362646, 5.281641628379971, 5.991905802775225, 6.348157066159785, 6.698977095541945, 7.277256315093645, 7.950163239439730, 8.321950569478621, 8.933205475999166, 9.507037814950760, 9.941326570560678, 10.30688674980187, 11.10979302513560, 11.34854823164315, 11.98547771694963, 12.18706923809562, 12.78473328130168, 13.23152491757729
