| L(s) = 1 | + 2-s + 7-s − 8-s + 9-s + 14-s − 16-s + 18-s − 19-s + 31-s − 38-s − 41-s − 2·47-s − 56-s − 59-s + 62-s + 63-s + 64-s − 2·67-s − 71-s − 72-s + 81-s − 82-s − 2·94-s + 97-s − 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 2-s + 7-s − 8-s + 9-s + 14-s − 16-s + 18-s − 19-s + 31-s − 38-s − 41-s − 2·47-s − 56-s − 59-s + 62-s + 63-s + 64-s − 2·67-s − 71-s − 72-s + 81-s − 82-s − 2·94-s + 97-s − 101-s + 103-s + 107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.536076732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536076732\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.60207216092712049351813564487, −9.752888595693298027823635203168, −8.720318221556186303223990781652, −7.967866030054034793429210542006, −6.83938356318854760055426335786, −5.96960166913563088305977798827, −4.74321072403891555671782504313, −4.48460650250333228109131475536, −3.24769481617007627267457235770, −1.77434508073061900636698426098,
1.77434508073061900636698426098, 3.24769481617007627267457235770, 4.48460650250333228109131475536, 4.74321072403891555671782504313, 5.96960166913563088305977798827, 6.83938356318854760055426335786, 7.967866030054034793429210542006, 8.720318221556186303223990781652, 9.752888595693298027823635203168, 10.60207216092712049351813564487
