| L(s) = 1 | + 2-s + 7-s − 8-s + 11-s + 14-s − 16-s + 22-s − 2·23-s + 25-s − 2·29-s − 37-s − 43-s − 2·46-s + 49-s + 50-s + 53-s − 56-s − 2·58-s + 64-s − 67-s + 71-s − 74-s + 77-s − 79-s − 86-s − 88-s + 98-s + ⋯ |
| L(s) = 1 | + 2-s + 7-s − 8-s + 11-s + 14-s − 16-s + 22-s − 2·23-s + 25-s − 2·29-s − 37-s − 43-s − 2·46-s + 49-s + 50-s + 53-s − 56-s − 2·58-s + 64-s − 67-s + 71-s − 74-s + 77-s − 79-s − 86-s − 88-s + 98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.389672557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389672557\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 + T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−11.32157696697024564129629293012, −10.14553842415203823536420191874, −9.099543907948821282974012311421, −8.391891963128610965800594435202, −7.24515281211028999737947655708, −6.12411644870711744305270421743, −5.29728918699943600080564234292, −4.31826818275458257861212228701, −3.56581216094107246068801523721, −1.91754799254750405515028727993,
1.91754799254750405515028727993, 3.56581216094107246068801523721, 4.31826818275458257861212228701, 5.29728918699943600080564234292, 6.12411644870711744305270421743, 7.24515281211028999737947655708, 8.391891963128610965800594435202, 9.099543907948821282974012311421, 10.14553842415203823536420191874, 11.32157696697024564129629293012
