| L(s) = 1 | − 1.21·2-s − 0.520·4-s − 1.24·5-s − 1.96·7-s + 3.06·8-s + 1.51·10-s + 1.16·11-s − 4.40·13-s + 2.38·14-s − 2.68·16-s + 1.37·17-s − 7.56·19-s + 0.648·20-s − 1.41·22-s − 8.73·23-s − 3.44·25-s + 5.35·26-s + 1.01·28-s − 6.85·29-s − 2.85·32-s − 1.67·34-s + 2.44·35-s + 7.41·37-s + 9.20·38-s − 3.82·40-s − 4.31·41-s − 5.62·43-s + ⋯ |
| L(s) = 1 | − 0.860·2-s − 0.260·4-s − 0.558·5-s − 0.741·7-s + 1.08·8-s + 0.480·10-s + 0.350·11-s − 1.22·13-s + 0.637·14-s − 0.672·16-s + 0.334·17-s − 1.73·19-s + 0.145·20-s − 0.301·22-s − 1.82·23-s − 0.688·25-s + 1.05·26-s + 0.192·28-s − 1.27·29-s − 0.505·32-s − 0.287·34-s + 0.413·35-s + 1.21·37-s + 1.49·38-s − 0.604·40-s − 0.673·41-s − 0.857·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.003253014093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003253014093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 5 | \( 1 + 1.24T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 - 1.16T + 11T^{2} \) |
| 13 | \( 1 + 4.40T + 13T^{2} \) |
| 17 | \( 1 - 1.37T + 17T^{2} \) |
| 19 | \( 1 + 7.56T + 19T^{2} \) |
| 23 | \( 1 + 8.73T + 23T^{2} \) |
| 29 | \( 1 + 6.85T + 29T^{2} \) |
| 37 | \( 1 - 7.41T + 37T^{2} \) |
| 41 | \( 1 + 4.31T + 41T^{2} \) |
| 43 | \( 1 + 5.62T + 43T^{2} \) |
| 47 | \( 1 - 4.48T + 47T^{2} \) |
| 53 | \( 1 + 1.29T + 53T^{2} \) |
| 59 | \( 1 + 0.539T + 59T^{2} \) |
| 61 | \( 1 + 7.98T + 61T^{2} \) |
| 67 | \( 1 + 4.71T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 7.78T + 79T^{2} \) |
| 83 | \( 1 + 2.80T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−7.81215713045884397085115704852, −7.39798511036182142106594816391, −6.48248411263293041380851283682, −5.88890760787529203952898933002, −4.83708978968530029948107492203, −4.15344243665256588641499661516, −3.66148124893451657278495747514, −2.42731684260151142493586250939, −1.64637290779651454855592427965, −0.03101500224883920959978658901,
0.03101500224883920959978658901, 1.64637290779651454855592427965, 2.42731684260151142493586250939, 3.66148124893451657278495747514, 4.15344243665256588641499661516, 4.83708978968530029948107492203, 5.88890760787529203952898933002, 6.48248411263293041380851283682, 7.39798511036182142106594816391, 7.81215713045884397085115704852
