| L(s) = 1 | + 2.61·2-s + 4.82·4-s − 4.07·5-s + 3.61·7-s + 7.39·8-s − 10.6·10-s + 11-s − 1.47·13-s + 9.45·14-s + 9.66·16-s + 3.27·17-s + 19-s − 19.6·20-s + 2.61·22-s + 7.45·23-s + 11.6·25-s − 3.86·26-s + 17.4·28-s − 1.02·29-s + 1.64·31-s + 10.4·32-s + 8.54·34-s − 14.7·35-s − 6.71·37-s + 2.61·38-s − 30.1·40-s + 3.92·41-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.41·4-s − 1.82·5-s + 1.36·7-s + 2.61·8-s − 3.36·10-s + 0.301·11-s − 0.410·13-s + 2.52·14-s + 2.41·16-s + 0.793·17-s + 0.229·19-s − 4.40·20-s + 0.557·22-s + 1.55·23-s + 2.32·25-s − 0.757·26-s + 3.30·28-s − 0.190·29-s + 0.296·31-s + 1.85·32-s + 1.46·34-s − 2.49·35-s − 1.10·37-s + 0.423·38-s − 4.76·40-s + 0.612·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.008402077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.008402077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 + 4.07T + 5T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 - 3.27T + 17T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 + 1.02T + 29T^{2} \) |
| 31 | \( 1 - 1.64T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 - 3.71T + 47T^{2} \) |
| 53 | \( 1 - 0.102T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 1.37T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 5.44T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.996621168830902758073874753867, −7.987391084397283949702801350945, −7.46753497005487261281183609159, −6.92604579839774860485591487784, −5.66298846613106537928684096284, −4.82585490054632765507610457510, −4.44093135603923078869639749746, −3.56537810870732679451359683539, −2.81663158388664963419337976073, −1.31593700445846289135337422907,
1.31593700445846289135337422907, 2.81663158388664963419337976073, 3.56537810870732679451359683539, 4.44093135603923078869639749746, 4.82585490054632765507610457510, 5.66298846613106537928684096284, 6.92604579839774860485591487784, 7.46753497005487261281183609159, 7.987391084397283949702801350945, 8.996621168830902758073874753867
