| L(s) = 1 | + 3-s − 3·5-s + 9-s − 4·11-s + 3·13-s − 3·15-s + 4·17-s − 23-s + 4·25-s + 27-s + 3·29-s − 6·31-s − 4·33-s − 9·37-s + 3·39-s − 9·41-s + 3·43-s − 3·45-s − 7·47-s + 4·51-s − 4·53-s + 12·55-s + 6·59-s − 10·61-s − 9·65-s − 4·67-s − 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s − 1.20·11-s + 0.832·13-s − 0.774·15-s + 0.970·17-s − 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.557·29-s − 1.07·31-s − 0.696·33-s − 1.47·37-s + 0.480·39-s − 1.40·41-s + 0.457·43-s − 0.447·45-s − 1.02·47-s + 0.560·51-s − 0.549·53-s + 1.61·55-s + 0.781·59-s − 1.28·61-s − 1.11·65-s − 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.207311592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207311592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.37784698740733, −14.04297859601922, −13.31871369031197, −12.99231293732292, −12.30984346565794, −11.99300461293201, −11.37675403281511, −10.79282212354095, −10.38939397468759, −9.865726047745802, −9.101170948462779, −8.465592804649598, −8.255992123142734, −7.561296304483026, −7.398764841354156, −6.578131427843676, −5.913736474362685, −5.079874222591639, −4.810924023827647, −3.801563019591936, −3.511761649608204, −3.059686706983326, −2.138678881055528, −1.376313573432241, −0.3739428911874132,
0.3739428911874132, 1.376313573432241, 2.138678881055528, 3.059686706983326, 3.511761649608204, 3.801563019591936, 4.810924023827647, 5.079874222591639, 5.913736474362685, 6.578131427843676, 7.398764841354156, 7.561296304483026, 8.255992123142734, 8.465592804649598, 9.101170948462779, 9.865726047745802, 10.38939397468759, 10.79282212354095, 11.37675403281511, 11.99300461293201, 12.30984346565794, 12.99231293732292, 13.31871369031197, 14.04297859601922, 14.37784698740733
