| L(s) = 1 | + 2.73·2-s + 1.50·3-s + 5.50·4-s − 1.91·5-s + 4.10·6-s − 0.472·7-s + 9.59·8-s − 0.748·9-s − 5.25·10-s − 2.34·11-s + 8.25·12-s + 0.319·13-s − 1.29·14-s − 2.87·15-s + 15.2·16-s − 0.228·17-s − 2.05·18-s + 3.36·19-s − 10.5·20-s − 0.708·21-s − 6.41·22-s + 23-s + 14.3·24-s − 1.32·25-s + 0.875·26-s − 5.62·27-s − 2.59·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.866·3-s + 2.75·4-s − 0.857·5-s + 1.67·6-s − 0.178·7-s + 3.39·8-s − 0.249·9-s − 1.66·10-s − 0.705·11-s + 2.38·12-s + 0.0886·13-s − 0.345·14-s − 0.742·15-s + 3.81·16-s − 0.0553·17-s − 0.483·18-s + 0.771·19-s − 2.35·20-s − 0.154·21-s − 1.36·22-s + 0.208·23-s + 2.93·24-s − 0.264·25-s + 0.171·26-s − 1.08·27-s − 0.490·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.133038303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.133038303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 + 1.91T + 5T^{2} \) |
| 7 | \( 1 + 0.472T + 7T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 - 0.319T + 13T^{2} \) |
| 17 | \( 1 + 0.228T + 17T^{2} \) |
| 19 | \( 1 - 3.36T + 19T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 + 5.23T + 37T^{2} \) |
| 41 | \( 1 - 1.97T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 6.50T + 47T^{2} \) |
| 53 | \( 1 + 0.128T + 53T^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 - 6.57T + 61T^{2} \) |
| 67 | \( 1 - 7.39T + 67T^{2} \) |
| 71 | \( 1 - 0.455T + 71T^{2} \) |
| 73 | \( 1 - 9.53T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 + 7.87T + 97T^{2} \) |
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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
−10.98566541650301871258636968362, −9.860636384021678485116021279018, −8.407707281569756405444689655820, −7.66209295947356250130756739450, −6.90141310785276288205349482918, −5.71186901073729439078643513588, −4.91372065645666418570032863168, −3.72316638466875045703674308372, −3.24081195882264538241593647355, −2.16706170665421223258469598115,
2.16706170665421223258469598115, 3.24081195882264538241593647355, 3.72316638466875045703674308372, 4.91372065645666418570032863168, 5.71186901073729439078643513588, 6.90141310785276288205349482918, 7.66209295947356250130756739450, 8.407707281569756405444689655820, 9.860636384021678485116021279018, 10.98566541650301871258636968362
