| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s − 16-s + 17-s − 19-s + 23-s + 25-s − 31-s + 34-s − 38-s + 40-s + 46-s − 2·47-s + 49-s + 50-s + 53-s − 61-s − 62-s + 64-s − 79-s + 80-s + 83-s − 85-s − 2·94-s + 95-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s − 8-s − 10-s − 16-s + 17-s − 19-s + 23-s + 25-s − 31-s + 34-s − 38-s + 40-s + 46-s − 2·47-s + 49-s + 50-s + 53-s − 61-s − 62-s + 64-s − 79-s + 80-s + 83-s − 85-s − 2·94-s + 95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7757333080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7757333080\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−13.35527806818721681265428230420, −12.56064074647946033816076711065, −11.78134711686748183415984780317, −10.70610207108181590025543040573, −9.210393751012387520528903635279, −8.145928751010362907909185854086, −6.84338781469083018794782030425, −5.43474497891861847503913399159, −4.29235474653493901087639687705, −3.20668977538890784336548404374,
3.20668977538890784336548404374, 4.29235474653493901087639687705, 5.43474497891861847503913399159, 6.84338781469083018794782030425, 8.145928751010362907909185854086, 9.210393751012387520528903635279, 10.70610207108181590025543040573, 11.78134711686748183415984780317, 12.56064074647946033816076711065, 13.35527806818721681265428230420
