| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 7-s − 8-s + 10-s + 11-s + 2·13-s − 14-s + 15-s − 16-s − 17-s − 21-s + 22-s + 2·23-s − 24-s + 2·26-s − 27-s + 29-s + 30-s + 33-s − 34-s − 35-s + 2·39-s − 40-s − 41-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 7-s − 8-s + 10-s + 11-s + 2·13-s − 14-s + 15-s − 16-s − 17-s − 21-s + 22-s + 2·23-s − 24-s + 2·26-s − 27-s + 29-s + 30-s + 33-s − 34-s − 35-s + 2·39-s − 40-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.882114173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.882114173\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( 1 - T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 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 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 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
−8.732079227984440401837668018739, −8.599648785757650607896723947816, −6.94956629369324294496604426995, −6.33181420128401282129949295514, −5.95363936057844479120414256932, −4.91523614124729509941377134678, −3.91698535557683382129868498929, −3.31983621538213056214662137172, −2.72933870611501332931821498792, −1.45849336512643250329147018009,
1.45849336512643250329147018009, 2.72933870611501332931821498792, 3.31983621538213056214662137172, 3.91698535557683382129868498929, 4.91523614124729509941377134678, 5.95363936057844479120414256932, 6.33181420128401282129949295514, 6.94956629369324294496604426995, 8.599648785757650607896723947816, 8.732079227984440401837668018739
