| L(s) = 1 | − 1.90·3-s − 44.5·5-s + 109.·7-s − 239.·9-s + 440.·11-s + 741.·13-s + 84.7·15-s − 1.56e3·17-s − 80.0·19-s − 208.·21-s + 381.·23-s − 1.14e3·25-s + 918.·27-s − 3.23e3·29-s − 4.07e3·31-s − 838.·33-s − 4.87e3·35-s − 3.29e3·37-s − 1.41e3·39-s + 1.11e4·41-s + 9.00e3·43-s + 1.06e4·45-s + 832.·47-s − 4.83e3·49-s + 2.98e3·51-s + 3.59e4·53-s − 1.96e4·55-s + ⋯ |
| L(s) = 1 | − 0.122·3-s − 0.796·5-s + 0.844·7-s − 0.985·9-s + 1.09·11-s + 1.21·13-s + 0.0972·15-s − 1.31·17-s − 0.0508·19-s − 0.103·21-s + 0.150·23-s − 0.365·25-s + 0.242·27-s − 0.714·29-s − 0.761·31-s − 0.134·33-s − 0.672·35-s − 0.395·37-s − 0.148·39-s + 1.03·41-s + 0.743·43-s + 0.784·45-s + 0.0549·47-s − 0.287·49-s + 0.160·51-s + 1.75·53-s − 0.873·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 1.90T + 243T^{2} \) |
| 5 | \( 1 + 44.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 109.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 440.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 741.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.56e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 80.0T + 2.47e6T^{2} \) |
| 23 | \( 1 - 381.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.29e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.11e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.00e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 832.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.83e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.46e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.04e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.16e5T + 8.58e9T^{2} \) |
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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
−8.705708499011775199509728746935, −8.134023902454955506641364559488, −7.12948424134616048403720219614, −6.23361671889216198667970063952, −5.38097686224928255188488031854, −4.18913516473296588799649808153, −3.68953713699802493211087285884, −2.28115782595500082069900215971, −1.15316054659929581587022317222, 0,
1.15316054659929581587022317222, 2.28115782595500082069900215971, 3.68953713699802493211087285884, 4.18913516473296588799649808153, 5.38097686224928255188488031854, 6.23361671889216198667970063952, 7.12948424134616048403720219614, 8.134023902454955506641364559488, 8.705708499011775199509728746935
