| L(s) = 1 | + 2.08·2-s + 3-s + 2.35·4-s − 2.36·5-s + 2.08·6-s − 0.791·7-s + 0.741·8-s + 9-s − 4.94·10-s + 1.46·11-s + 2.35·12-s − 2.45·13-s − 1.65·14-s − 2.36·15-s − 3.16·16-s + 17-s + 2.08·18-s − 0.298·19-s − 5.57·20-s − 0.791·21-s + 3.04·22-s − 7.61·23-s + 0.741·24-s + 0.609·25-s − 5.13·26-s + 27-s − 1.86·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.577·3-s + 1.17·4-s − 1.05·5-s + 0.851·6-s − 0.299·7-s + 0.262·8-s + 0.333·9-s − 1.56·10-s + 0.440·11-s + 0.679·12-s − 0.681·13-s − 0.441·14-s − 0.611·15-s − 0.790·16-s + 0.242·17-s + 0.491·18-s − 0.0683·19-s − 1.24·20-s − 0.172·21-s + 0.649·22-s − 1.58·23-s + 0.151·24-s + 0.121·25-s − 1.00·26-s + 0.192·27-s − 0.352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 + 0.791T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 2.45T + 13T^{2} \) |
| 19 | \( 1 + 0.298T + 19T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 31 | \( 1 + 2.70T + 31T^{2} \) |
| 37 | \( 1 - 9.81T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 + 8.90T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 0.660T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 1.42T + 73T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 + 1.32T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.936339148613568100595035437177, −7.24111397890786002731510559280, −6.51850297826681062279972530254, −5.73938374412029982678059471308, −4.82338398143170228155152302753, −4.10340879050797746475039170035, −3.64814666970003659269426111998, −2.88471734001765507164040013617, −1.89533896563017965080689809975, 0,
1.89533896563017965080689809975, 2.88471734001765507164040013617, 3.64814666970003659269426111998, 4.10340879050797746475039170035, 4.82338398143170228155152302753, 5.73938374412029982678059471308, 6.51850297826681062279972530254, 7.24111397890786002731510559280, 7.936339148613568100595035437177
