| L(s) = 1 | − 3.92·2-s + 9·3-s − 16.5·4-s − 77.2·5-s − 35.3·6-s − 239.·7-s + 190.·8-s + 81·9-s + 303.·10-s + 190.·11-s − 149.·12-s − 743.·13-s + 940.·14-s − 695.·15-s − 219.·16-s − 318.·18-s + 2.04e3·19-s + 1.27e3·20-s − 2.15e3·21-s − 749.·22-s − 3.45e3·23-s + 1.71e3·24-s + 2.84e3·25-s + 2.92e3·26-s + 729·27-s + 3.96e3·28-s − 926.·29-s + ⋯ |
| L(s) = 1 | − 0.694·2-s + 0.577·3-s − 0.517·4-s − 1.38·5-s − 0.401·6-s − 1.84·7-s + 1.05·8-s + 0.333·9-s + 0.960·10-s + 0.475·11-s − 0.298·12-s − 1.22·13-s + 1.28·14-s − 0.798·15-s − 0.214·16-s − 0.231·18-s + 1.30·19-s + 0.715·20-s − 1.06·21-s − 0.330·22-s − 1.36·23-s + 0.608·24-s + 0.911·25-s + 0.847·26-s + 0.192·27-s + 0.955·28-s − 0.204·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 3.92T + 32T^{2} \) |
| 5 | \( 1 + 77.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 239.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 190.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 743.T + 3.71e5T^{2} \) |
| 19 | \( 1 - 2.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 926.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.12e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.98e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.28e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.32e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.78e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.12e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.99e4T + 8.58e9T^{2} \) |
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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
−9.137735434506137334798750656424, −8.157706095990561107089333141460, −7.44233101067192605425772752459, −6.88527629768781342509256862511, −5.43778880354018981549158601032, −4.03109932015753691038478039205, −3.71339427128015327947294393370, −2.53557567863186622830918924567, −0.76711566208416153418776748461, 0,
0.76711566208416153418776748461, 2.53557567863186622830918924567, 3.71339427128015327947294393370, 4.03109932015753691038478039205, 5.43778880354018981549158601032, 6.88527629768781342509256862511, 7.44233101067192605425772752459, 8.157706095990561107089333141460, 9.137735434506137334798750656424
