| L(s) = 1 | + 1.18·5-s + 7-s + 3.70·11-s + 13-s − 6.94·17-s − 1.94·19-s + 5.60·23-s − 3.60·25-s + 0.239·29-s − 1.66·31-s + 1.18·35-s − 9.54·37-s − 10.1·41-s − 2.22·43-s − 5.82·47-s + 49-s − 11.6·53-s + 4.37·55-s − 2.60·59-s − 7.60·61-s + 1.18·65-s − 3.50·67-s − 8.60·71-s + 15.1·73-s + 3.70·77-s − 7.37·79-s + 6.94·83-s + ⋯ |
| L(s) = 1 | + 0.528·5-s + 0.377·7-s + 1.11·11-s + 0.277·13-s − 1.68·17-s − 0.445·19-s + 1.16·23-s − 0.720·25-s + 0.0444·29-s − 0.298·31-s + 0.199·35-s − 1.56·37-s − 1.59·41-s − 0.339·43-s − 0.850·47-s + 0.142·49-s − 1.59·53-s + 0.590·55-s − 0.338·59-s − 0.973·61-s + 0.146·65-s − 0.428·67-s − 1.02·71-s + 1.77·73-s + 0.422·77-s − 0.830·79-s + 0.762·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 1.18T + 5T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 - 5.60T + 23T^{2} \) |
| 29 | \( 1 - 0.239T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 + 9.54T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.22T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 + 8.60T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 7.37T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.23516457147933117641356212681, −6.56756885795437889178425744036, −6.27378151503596418886592598481, −5.20339155961942911440644235514, −4.67899439831853684315854346705, −3.86523596296885761171498942116, −3.08882304553619721310908113002, −1.93340850230431137953131829869, −1.51665031095522094948295371284, 0,
1.51665031095522094948295371284, 1.93340850230431137953131829869, 3.08882304553619721310908113002, 3.86523596296885761171498942116, 4.67899439831853684315854346705, 5.20339155961942911440644235514, 6.27378151503596418886592598481, 6.56756885795437889178425744036, 7.23516457147933117641356212681
