| L(s) = 1 | + 2-s − 3-s − 6-s − 7-s − 8-s + 9-s + 11-s − 13-s − 14-s − 16-s + 17-s + 18-s + 21-s + 22-s + 24-s + 25-s − 26-s − 27-s − 29-s − 33-s + 34-s + 39-s − 2·41-s + 42-s + 47-s + 48-s + 50-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 6-s − 7-s − 8-s + 9-s + 11-s − 13-s − 14-s − 16-s + 17-s + 18-s + 21-s + 22-s + 24-s + 25-s − 26-s − 27-s − 29-s − 33-s + 34-s + 39-s − 2·41-s + 42-s + 47-s + 48-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5770586370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5770586370\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 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 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−14.39892195706409998885986607529, −13.18133367873374302632506132995, −12.37546425848442860075315832869, −11.74152542706301551240275566622, −10.16119846875737190734833337722, −9.215490822664834597853334075143, −7.06152498257571751488998986929, −6.04035672910010770144714336001, −4.91310283918857793479423733862, −3.53215773738360323132247593606,
3.53215773738360323132247593606, 4.91310283918857793479423733862, 6.04035672910010770144714336001, 7.06152498257571751488998986929, 9.215490822664834597853334075143, 10.16119846875737190734833337722, 11.74152542706301551240275566622, 12.37546425848442860075315832869, 13.18133367873374302632506132995, 14.39892195706409998885986607529
