| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 11-s − 15-s − 2·17-s − 4·19-s − 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s − 33-s + 4·35-s − 10·37-s + 10·41-s + 4·43-s + 45-s + 9·49-s + 2·51-s + 10·53-s + 55-s + 4·57-s + 12·59-s − 2·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.174·33-s + 0.676·35-s − 1.64·37-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s + 0.280·51-s + 1.37·53-s + 0.134·55-s + 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 223080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 223080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.994444220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.994444220\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.95081415483187, −12.27553974888704, −12.03913002987728, −11.63725067644384, −10.95520614867101, −10.72568088105151, −10.37029233915834, −9.745200353733788, −9.179492807117804, −8.577920750803719, −8.377135673673863, −7.775476375478920, −7.153356355876493, −6.779366473007443, −6.144425589252708, −5.742374001373886, −5.175848671065721, −4.768335943650553, −4.115650340375349, −3.962481024145832, −2.858742831370400, −2.198594708321206, −1.861032748530301, −1.161743971076424, −0.5285163344620151,
0.5285163344620151, 1.161743971076424, 1.861032748530301, 2.198594708321206, 2.858742831370400, 3.962481024145832, 4.115650340375349, 4.768335943650553, 5.175848671065721, 5.742374001373886, 6.144425589252708, 6.779366473007443, 7.153356355876493, 7.775476375478920, 8.377135673673863, 8.577920750803719, 9.179492807117804, 9.745200353733788, 10.37029233915834, 10.72568088105151, 10.95520614867101, 11.63725067644384, 12.03913002987728, 12.27553974888704, 12.95081415483187
