| L(s) = 1 | − 0.972·2-s − 1.50·3-s − 1.05·4-s − 2.83·5-s + 1.46·6-s + 2.96·8-s − 0.730·9-s + 2.75·10-s + 3.37·11-s + 1.58·12-s − 6.81·13-s + 4.27·15-s − 0.776·16-s − 0.331·17-s + 0.709·18-s − 8.31·19-s + 2.99·20-s − 3.27·22-s − 5.67·23-s − 4.47·24-s + 3.04·25-s + 6.62·26-s + 5.61·27-s − 1.34·29-s − 4.15·30-s − 0.541·31-s − 5.18·32-s + ⋯ |
| L(s) = 1 | − 0.687·2-s − 0.869·3-s − 0.527·4-s − 1.26·5-s + 0.597·6-s + 1.04·8-s − 0.243·9-s + 0.871·10-s + 1.01·11-s + 0.458·12-s − 1.88·13-s + 1.10·15-s − 0.194·16-s − 0.0804·17-s + 0.167·18-s − 1.90·19-s + 0.669·20-s − 0.698·22-s − 1.18·23-s − 0.913·24-s + 0.608·25-s + 1.29·26-s + 1.08·27-s − 0.249·29-s − 0.758·30-s − 0.0972·31-s − 0.916·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02409618912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02409618912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.972T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 + 2.83T + 5T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 13 | \( 1 + 6.81T + 13T^{2} \) |
| 17 | \( 1 + 0.331T + 17T^{2} \) |
| 19 | \( 1 + 8.31T + 19T^{2} \) |
| 23 | \( 1 + 5.67T + 23T^{2} \) |
| 29 | \( 1 + 1.34T + 29T^{2} \) |
| 31 | \( 1 + 0.541T + 31T^{2} \) |
| 37 | \( 1 + 4.20T + 37T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 + 4.07T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 61 | \( 1 - 0.720T + 61T^{2} \) |
| 67 | \( 1 + 4.26T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 8.47T + 79T^{2} \) |
| 83 | \( 1 - 7.88T + 83T^{2} \) |
| 89 | \( 1 - 2.31T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−9.065043479378161026453351495677, −8.372637852631086223279016701311, −7.73784714457847535593178576515, −6.92235918590899393424046735388, −6.11922743779747619483105484476, −4.82008867106540094570270279772, −4.50402236506919979027284766538, −3.49882116172296903556667941825, −1.88479496551374640114966162867, −0.11566891612279901486891310766,
0.11566891612279901486891310766, 1.88479496551374640114966162867, 3.49882116172296903556667941825, 4.50402236506919979027284766538, 4.82008867106540094570270279772, 6.11922743779747619483105484476, 6.92235918590899393424046735388, 7.73784714457847535593178576515, 8.372637852631086223279016701311, 9.065043479378161026453351495677
