L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 13-s − 15-s + 6·17-s + 4·21-s − 8·23-s + 25-s + 27-s − 6·29-s − 4·31-s − 4·35-s − 2·37-s − 39-s + 10·41-s + 4·43-s − 45-s − 8·47-s + 9·49-s + 6·51-s − 2·53-s + 12·59-s + 2·61-s + 4·63-s + 65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 1.45·17-s + 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.676·35-s − 0.328·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s + 1.56·59-s + 0.256·61-s + 0.503·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.545756746\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.545756746\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.12846101093221, −12.56397598688996, −12.10255495821785, −11.81989273870952, −11.19811832179551, −10.85421020063281, −10.31884441267507, −9.670952202239712, −9.442625053746943, −8.655251806551797, −8.250702867898641, −7.894068622701575, −7.419343314727319, −7.247969758424682, −6.242745262073713, −5.674447563410418, −5.334083976475302, −4.623636259847702, −4.190825855649476, −3.683567973328526, −3.153194783228843, −2.323780871602032, −1.864899829516904, −1.331446325880176, −0.5165951997112039,
0.5165951997112039, 1.331446325880176, 1.864899829516904, 2.323780871602032, 3.153194783228843, 3.683567973328526, 4.190825855649476, 4.623636259847702, 5.334083976475302, 5.674447563410418, 6.242745262073713, 7.247969758424682, 7.419343314727319, 7.894068622701575, 8.250702867898641, 8.655251806551797, 9.442625053746943, 9.670952202239712, 10.31884441267507, 10.85421020063281, 11.19811832179551, 11.81989273870952, 12.10255495821785, 12.56397598688996, 13.12846101093221