| L(s) = 1 | + 1.46·2-s − 2.50·3-s + 0.157·4-s + 4.31·5-s − 3.67·6-s + 1.97·7-s − 2.70·8-s + 3.26·9-s + 6.34·10-s + 1.83·11-s − 0.393·12-s − 3.87·13-s + 2.90·14-s − 10.8·15-s − 4.28·16-s + 4.20·17-s + 4.78·18-s + 5.07·19-s + 0.679·20-s − 4.94·21-s + 2.69·22-s − 23-s + 6.77·24-s + 13.6·25-s − 5.68·26-s − 0.650·27-s + 0.310·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 1.44·3-s + 0.0786·4-s + 1.93·5-s − 1.50·6-s + 0.747·7-s − 0.956·8-s + 1.08·9-s + 2.00·10-s + 0.552·11-s − 0.113·12-s − 1.07·13-s + 0.776·14-s − 2.78·15-s − 1.07·16-s + 1.01·17-s + 1.12·18-s + 1.16·19-s + 0.151·20-s − 1.07·21-s + 0.574·22-s − 0.208·23-s + 1.38·24-s + 2.72·25-s − 1.11·26-s − 0.125·27-s + 0.0587·28-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
| \(L(1)\) |
\(\approx\) |
\(2.103281051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103281051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.46T + 2T^{2} \) |
| 3 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 - 4.31T + 5T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 - 9.06T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 + 6.05T + 47T^{2} \) |
| 53 | \( 1 - 2.15T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 + 8.07T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 + 2.02T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 + 2.99T + 89T^{2} \) |
| 97 | \( 1 + 2.79T + 97T^{2} \) |
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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
−10.67399338836195249335732621058, −9.660668259157267071368755927016, −9.286356382478117056811896627607, −7.52965645731244269809838810329, −6.31625248523428475197186371759, −5.78643535503820794644389159292, −5.20338884624992411243223656890, −4.55412687241821955396993317857, −2.79440050873709987585568695825, −1.30500374845837214906970266553,
1.30500374845837214906970266553, 2.79440050873709987585568695825, 4.55412687241821955396993317857, 5.20338884624992411243223656890, 5.78643535503820794644389159292, 6.31625248523428475197186371759, 7.52965645731244269809838810329, 9.286356382478117056811896627607, 9.660668259157267071368755927016, 10.67399338836195249335732621058
