L(s) = 1 | − 1.22·2-s + 3-s − 0.507·4-s + 0.145·5-s − 1.22·6-s + 3.06·8-s + 9-s − 0.177·10-s + 1.83·11-s − 0.507·12-s − 2.53·13-s + 0.145·15-s − 2.72·16-s − 5.73·17-s − 1.22·18-s − 5.97·19-s − 0.0738·20-s − 2.23·22-s − 7.46·23-s + 3.06·24-s − 4.97·25-s + 3.09·26-s + 27-s − 0.614·29-s − 0.177·30-s + 9.81·31-s − 2.79·32-s + ⋯ |
L(s) = 1 | − 0.863·2-s + 0.577·3-s − 0.253·4-s + 0.0650·5-s − 0.498·6-s + 1.08·8-s + 0.333·9-s − 0.0561·10-s + 0.552·11-s − 0.146·12-s − 0.702·13-s + 0.0375·15-s − 0.681·16-s − 1.39·17-s − 0.287·18-s − 1.37·19-s − 0.0165·20-s − 0.477·22-s − 1.55·23-s + 0.625·24-s − 0.995·25-s + 0.606·26-s + 0.192·27-s − 0.114·29-s − 0.0324·30-s + 1.76·31-s − 0.494·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9442446892\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9442446892\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 5 | \( 1 - 0.145T + 5T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 + 7.46T + 23T^{2} \) |
| 29 | \( 1 + 0.614T + 29T^{2} \) |
| 31 | \( 1 - 9.81T + 31T^{2} \) |
| 37 | \( 1 + 1.76T + 37T^{2} \) |
| 43 | \( 1 - 2.38T + 43T^{2} \) |
| 47 | \( 1 - 8.27T + 47T^{2} \) |
| 53 | \( 1 + 6.25T + 53T^{2} \) |
| 59 | \( 1 - 8.36T + 59T^{2} \) |
| 61 | \( 1 - 2.56T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 - 7.02T + 79T^{2} \) |
| 83 | \( 1 + 1.70T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 6.11T + 97T^{2} \) |
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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
−8.219312834988219543678212051491, −7.68241098985903195460409696751, −6.74592033853144013006447287955, −6.25130743838285389259388695349, −5.03933778065614961083091839473, −4.23259338490977586769469045724, −3.88592277723489814565335683119, −2.33990906099518597302816217572, −1.96462015304087928909311955794, −0.55192210315641414423167203250,
0.55192210315641414423167203250, 1.96462015304087928909311955794, 2.33990906099518597302816217572, 3.88592277723489814565335683119, 4.23259338490977586769469045724, 5.03933778065614961083091839473, 6.25130743838285389259388695349, 6.74592033853144013006447287955, 7.68241098985903195460409696751, 8.219312834988219543678212051491