L(s) = 1 | − 2.41·2-s + 2.60·3-s + 3.83·4-s + 1.90·5-s − 6.30·6-s − 0.0638·7-s − 4.43·8-s + 3.80·9-s − 4.60·10-s − 0.824·11-s + 10.0·12-s + 13-s + 0.154·14-s + 4.97·15-s + 3.03·16-s + 1.70·17-s − 9.19·18-s − 3.21·19-s + 7.30·20-s − 0.166·21-s + 1.99·22-s − 7.15·23-s − 11.5·24-s − 1.36·25-s − 2.41·26-s + 2.10·27-s − 0.244·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.50·3-s + 1.91·4-s + 0.852·5-s − 2.57·6-s − 0.0241·7-s − 1.56·8-s + 1.26·9-s − 1.45·10-s − 0.248·11-s + 2.88·12-s + 0.277·13-s + 0.0411·14-s + 1.28·15-s + 0.759·16-s + 0.412·17-s − 2.16·18-s − 0.738·19-s + 1.63·20-s − 0.0363·21-s + 0.424·22-s − 1.49·23-s − 2.36·24-s − 0.273·25-s − 0.473·26-s + 0.404·27-s − 0.0462·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 1.90T + 5T^{2} \) |
| 7 | \( 1 + 0.0638T + 7T^{2} \) |
| 11 | \( 1 + 0.824T + 11T^{2} \) |
| 17 | \( 1 - 1.70T + 17T^{2} \) |
| 19 | \( 1 + 3.21T + 19T^{2} \) |
| 23 | \( 1 + 7.15T + 23T^{2} \) |
| 29 | \( 1 + 9.58T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 8.31T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 - 7.96T + 79T^{2} \) |
| 83 | \( 1 + 5.46T + 83T^{2} \) |
| 89 | \( 1 - 7.57T + 89T^{2} \) |
| 97 | \( 1 - 5.99T + 97T^{2} \) |
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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
−7.897647821607180319479539426319, −7.28696637706899184027705446671, −6.24803346383001923453364375732, −5.84966367388261512495161765009, −4.45886450740011223417786294135, −3.53147031616875455004862272557, −2.68426657552024440321008450110, −1.93921134312420866654196472728, −1.58078240402423364460040398365, 0,
1.58078240402423364460040398365, 1.93921134312420866654196472728, 2.68426657552024440321008450110, 3.53147031616875455004862272557, 4.45886450740011223417786294135, 5.84966367388261512495161765009, 6.24803346383001923453364375732, 7.28696637706899184027705446671, 7.897647821607180319479539426319