| L(s) = 1 | − 2.05·2-s − 1.62·3-s + 2.20·4-s + 4.18·5-s + 3.34·6-s − 7-s − 0.421·8-s − 0.345·9-s − 8.57·10-s − 3.88·11-s − 3.59·12-s + 2.89·13-s + 2.05·14-s − 6.81·15-s − 3.54·16-s + 2.83·17-s + 0.707·18-s + 5.96·19-s + 9.21·20-s + 1.62·21-s + 7.96·22-s + 2.37·23-s + 0.686·24-s + 12.4·25-s − 5.94·26-s + 5.45·27-s − 2.20·28-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.940·3-s + 1.10·4-s + 1.86·5-s + 1.36·6-s − 0.377·7-s − 0.148·8-s − 0.115·9-s − 2.71·10-s − 1.17·11-s − 1.03·12-s + 0.804·13-s + 0.548·14-s − 1.75·15-s − 0.886·16-s + 0.687·17-s + 0.166·18-s + 1.36·19-s + 2.06·20-s + 0.355·21-s + 1.69·22-s + 0.494·23-s + 0.140·24-s + 2.49·25-s − 1.16·26-s + 1.04·27-s − 0.416·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5911170755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5911170755\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 3 | \( 1 + 1.62T + 3T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 19 | \( 1 - 5.96T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + 3.48T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 43 | \( 1 - 5.56T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 - 1.15T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 0.0168T + 73T^{2} \) |
| 79 | \( 1 + 0.301T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−11.21048271750631073248132393502, −10.73109327081354778579170674672, −9.743630404675899566911603655793, −9.393819041811109270413037100347, −8.138958378120491389877987356009, −6.94111091657303360575520922579, −5.83337267646263694445845412637, −5.31807492551881003216459297810, −2.64553041625941102682902178844, −1.09360493080290017379862816227,
1.09360493080290017379862816227, 2.64553041625941102682902178844, 5.31807492551881003216459297810, 5.83337267646263694445845412637, 6.94111091657303360575520922579, 8.138958378120491389877987356009, 9.393819041811109270413037100347, 9.743630404675899566911603655793, 10.73109327081354778579170674672, 11.21048271750631073248132393502
