| L(s) = 1 | + 8.54·5-s + 8.54·7-s + 11-s + 8·13-s + 102.·17-s + 119.·19-s + 18·23-s − 52.0·25-s − 119.·29-s + 93.9·31-s + 72.9·35-s + 146·37-s − 85.4·41-s − 222.·43-s + 106·47-s − 270·49-s + 299.·53-s + 8.54·55-s + 20·59-s + 408·61-s + 68.3·65-s + 598.·67-s − 20·71-s − 591·73-s + 8.54·77-s + 683.·79-s + 345·83-s + ⋯ |
| L(s) = 1 | + 0.764·5-s + 0.461·7-s + 0.0274·11-s + 0.170·13-s + 1.46·17-s + 1.44·19-s + 0.163·23-s − 0.416·25-s − 0.765·29-s + 0.544·31-s + 0.352·35-s + 0.648·37-s − 0.325·41-s − 0.787·43-s + 0.328·47-s − 0.787·49-s + 0.775·53-s + 0.0209·55-s + 0.0441·59-s + 0.856·61-s + 0.130·65-s + 1.09·67-s − 0.0334·71-s − 0.947·73-s + 0.0126·77-s + 0.973·79-s + 0.456·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.166650363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.166650363\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 8.54T + 125T^{2} \) |
| 7 | \( 1 - 8.54T + 343T^{2} \) |
| 11 | \( 1 - T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18T + 1.21e4T^{2} \) |
| 29 | \( 1 + 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 93.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 146T + 5.06e4T^{2} \) |
| 41 | \( 1 + 85.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 222.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 106T + 1.03e5T^{2} \) |
| 53 | \( 1 - 299.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 20T + 2.05e5T^{2} \) |
| 61 | \( 1 - 408T + 2.26e5T^{2} \) |
| 67 | \( 1 - 598.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 20T + 3.57e5T^{2} \) |
| 73 | \( 1 + 591T + 3.89e5T^{2} \) |
| 79 | \( 1 - 683.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 345T + 5.71e5T^{2} \) |
| 89 | \( 1 + 222.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.24e3T + 9.12e5T^{2} \) |
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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
−9.076711312530352854120303407241, −8.053354183754915996755404035190, −7.51361326868520746330958371068, −6.48986362485961314301649082726, −5.55657509747208768155164280030, −5.14736816795559432923617116088, −3.85648373260802052324975502822, −2.94642202039581870532625287037, −1.77910959052758996853578676943, −0.900721613741799595041368723147,
0.900721613741799595041368723147, 1.77910959052758996853578676943, 2.94642202039581870532625287037, 3.85648373260802052324975502822, 5.14736816795559432923617116088, 5.55657509747208768155164280030, 6.48986362485961314301649082726, 7.51361326868520746330958371068, 8.053354183754915996755404035190, 9.076711312530352854120303407241
