| L(s) = 1 | − 1.39·2-s − 2.49·3-s − 0.0539·4-s − 3.75·5-s + 3.47·6-s − 3.19·7-s + 2.86·8-s + 3.21·9-s + 5.23·10-s + 3.83·11-s + 0.134·12-s + 4.68·13-s + 4.45·14-s + 9.35·15-s − 3.88·16-s + 17-s − 4.48·18-s − 4.04·19-s + 0.202·20-s + 7.96·21-s − 5.34·22-s + 2.89·23-s − 7.14·24-s + 9.08·25-s − 6.53·26-s − 0.540·27-s + 0.172·28-s + ⋯ |
| L(s) = 1 | − 0.986·2-s − 1.43·3-s − 0.0269·4-s − 1.67·5-s + 1.42·6-s − 1.20·7-s + 1.01·8-s + 1.07·9-s + 1.65·10-s + 1.15·11-s + 0.0388·12-s + 1.29·13-s + 1.19·14-s + 2.41·15-s − 0.972·16-s + 0.242·17-s − 1.05·18-s − 0.927·19-s + 0.0452·20-s + 1.73·21-s − 1.13·22-s + 0.602·23-s − 1.45·24-s + 1.81·25-s − 1.28·26-s − 0.104·27-s + 0.0325·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.39T + 2T^{2} \) |
| 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 - 2.89T + 23T^{2} \) |
| 29 | \( 1 + 9.80T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 + 4.11T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 1.11T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 0.882T + 67T^{2} \) |
| 71 | \( 1 + 2.76T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 8.70T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−9.911230286285675098678839577167, −8.914567207066573516819017770529, −8.331057509840042813526989910243, −7.06616019257612257170439874695, −6.68204988835670210367529806414, −5.52916560079428828321099190115, −4.15364194312270436858233944419, −3.70136500574671525036330156776, −1.01038489849111116743966586150, 0,
1.01038489849111116743966586150, 3.70136500574671525036330156776, 4.15364194312270436858233944419, 5.52916560079428828321099190115, 6.68204988835670210367529806414, 7.06616019257612257170439874695, 8.331057509840042813526989910243, 8.914567207066573516819017770529, 9.911230286285675098678839577167
