| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 16-s + 18-s − 23-s − 24-s + 25-s − 27-s − 2·29-s + 32-s + 36-s − 46-s − 2·47-s − 48-s − 49-s + 50-s − 54-s − 2·58-s + 64-s + 69-s + 2·71-s + 72-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 16-s + 18-s − 23-s − 24-s + 25-s − 27-s − 2·29-s + 32-s + 36-s − 46-s − 2·47-s − 48-s − 49-s + 50-s − 54-s − 2·58-s + 64-s + 69-s + 2·71-s + 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.243473209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243473209\) |
| \(L(1)\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−11.25089423817438257865029743072, −10.44794253854474455875359048298, −9.551699880810596791363138125413, −8.004172508750204469142437841891, −7.06723073766203644294168139506, −6.24391935809895660806246498302, −5.41246971218832358330532844747, −4.54445018999764363495254600017, −3.47604328454255466387141630995, −1.80871561547600075308868836756,
1.80871561547600075308868836756, 3.47604328454255466387141630995, 4.54445018999764363495254600017, 5.41246971218832358330532844747, 6.24391935809895660806246498302, 7.06723073766203644294168139506, 8.004172508750204469142437841891, 9.551699880810596791363138125413, 10.44794253854474455875359048298, 11.25089423817438257865029743072
