| L(s) = 1 | − 2.66·2-s + 0.632·3-s + 5.09·4-s − 0.0737·5-s − 1.68·6-s + 1.42·7-s − 8.25·8-s − 2.59·9-s + 0.196·10-s + 11-s + 3.22·12-s − 1.48·13-s − 3.79·14-s − 0.0466·15-s + 11.8·16-s + 5.01·17-s + 6.92·18-s + 4.10·19-s − 0.375·20-s + 0.902·21-s − 2.66·22-s + 4.32·23-s − 5.22·24-s − 4.99·25-s + 3.95·26-s − 3.54·27-s + 7.26·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 0.365·3-s + 2.54·4-s − 0.0329·5-s − 0.688·6-s + 0.538·7-s − 2.91·8-s − 0.866·9-s + 0.0621·10-s + 0.301·11-s + 0.931·12-s − 0.411·13-s − 1.01·14-s − 0.0120·15-s + 2.95·16-s + 1.21·17-s + 1.63·18-s + 0.940·19-s − 0.0840·20-s + 0.196·21-s − 0.568·22-s + 0.902·23-s − 1.06·24-s − 0.998·25-s + 0.775·26-s − 0.682·27-s + 1.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 - 0.632T + 3T^{2} \) |
| 5 | \( 1 + 0.0737T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 - 4.32T + 23T^{2} \) |
| 29 | \( 1 + 3.98T + 29T^{2} \) |
| 31 | \( 1 + 5.93T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 + 1.93T + 41T^{2} \) |
| 43 | \( 1 + 7.31T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 0.702T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 + 1.72T + 79T^{2} \) |
| 83 | \( 1 + 1.11T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−7.75750301159717866285700407537, −7.55270100497452777409755757627, −6.63202169485033371549432877958, −5.76527387686466230904579180972, −5.13982651164883799833567847549, −3.57220309680354908576303015156, −2.95147574455863377456336152155, −1.97491123188144996193091614142, −1.22286948586938958916151036524, 0,
1.22286948586938958916151036524, 1.97491123188144996193091614142, 2.95147574455863377456336152155, 3.57220309680354908576303015156, 5.13982651164883799833567847549, 5.76527387686466230904579180972, 6.63202169485033371549432877958, 7.55270100497452777409755757627, 7.75750301159717866285700407537
