| L(s) = 1 | − 2.37·2-s + 0.630·3-s + 3.62·4-s + 1.61·5-s − 1.49·6-s − 3.60·7-s − 3.85·8-s − 2.60·9-s − 3.83·10-s + 3.25·11-s + 2.28·12-s + 6.29·13-s + 8.55·14-s + 1.01·15-s + 1.88·16-s − 2.89·17-s + 6.17·18-s + 2.32·19-s + 5.85·20-s − 2.27·21-s − 7.72·22-s + 2.72·23-s − 2.43·24-s − 2.39·25-s − 14.9·26-s − 3.53·27-s − 13.0·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.364·3-s + 1.81·4-s + 0.722·5-s − 0.610·6-s − 1.36·7-s − 1.36·8-s − 0.867·9-s − 1.21·10-s + 0.981·11-s + 0.660·12-s + 1.74·13-s + 2.28·14-s + 0.263·15-s + 0.471·16-s − 0.702·17-s + 1.45·18-s + 0.533·19-s + 1.30·20-s − 0.496·21-s − 1.64·22-s + 0.567·23-s − 0.496·24-s − 0.478·25-s − 2.92·26-s − 0.680·27-s − 2.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8430132176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8430132176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 - 0.630T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + 7.05T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 - 0.106T + 37T^{2} \) |
| 41 | \( 1 - 8.84T + 41T^{2} \) |
| 47 | \( 1 + 0.939T + 47T^{2} \) |
| 53 | \( 1 - 2.33T + 53T^{2} \) |
| 59 | \( 1 - 6.07T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 5.21T + 71T^{2} \) |
| 73 | \( 1 - 4.61T + 73T^{2} \) |
| 79 | \( 1 + 1.77T + 79T^{2} \) |
| 83 | \( 1 - 6.67T + 83T^{2} \) |
| 89 | \( 1 - 2.38T + 89T^{2} \) |
| 97 | \( 1 - 7.44T + 97T^{2} \) |
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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
−9.193416367170610729218513675695, −8.819443961995886673611041070904, −7.967767980511374543977492517706, −6.92737838583775531730567067185, −6.27355879655734491304892312072, −5.80109621661301983034073128051, −3.88803291346753700823545956698, −3.00744386051549819182958405361, −1.96335291972495678796955309377, −0.78827954107093305543494942803,
0.78827954107093305543494942803, 1.96335291972495678796955309377, 3.00744386051549819182958405361, 3.88803291346753700823545956698, 5.80109621661301983034073128051, 6.27355879655734491304892312072, 6.92737838583775531730567067185, 7.967767980511374543977492517706, 8.819443961995886673611041070904, 9.193416367170610729218513675695
