| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 11-s − 12-s + 16-s − 2·17-s + 18-s − 22-s − 24-s − 25-s − 27-s + 32-s + 33-s − 2·34-s + 36-s + 2·41-s − 44-s − 48-s − 49-s − 50-s + 2·51-s − 54-s + 64-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 11-s − 12-s + 16-s − 2·17-s + 18-s − 22-s − 24-s − 25-s − 27-s + 32-s + 33-s − 2·34-s + 36-s + 2·41-s − 44-s − 48-s − 49-s − 50-s + 2·51-s − 54-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9772441666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9772441666\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−12.29770412392481430846900693204, −11.18468460690600884282203649779, −10.85333902973642863237342355320, −9.649510736362953362776784017192, −7.960507410695990802463502940914, −6.89894249237026399844596740231, −6.01870258429294678517792255106, −5.01783247758821344530543464480, −4.09430079775723958544645069627, −2.27898768068874343474384428778,
2.27898768068874343474384428778, 4.09430079775723958544645069627, 5.01783247758821344530543464480, 6.01870258429294678517792255106, 6.89894249237026399844596740231, 7.960507410695990802463502940914, 9.649510736362953362776784017192, 10.85333902973642863237342355320, 11.18468460690600884282203649779, 12.29770412392481430846900693204
