| L(s) = 1 | − 0.779·2-s − 3-s − 1.39·4-s − 2.98·5-s + 0.779·6-s − 2.49·7-s + 2.64·8-s + 9-s + 2.32·10-s + 1.94·11-s + 1.39·12-s − 3.49·13-s + 1.94·14-s + 2.98·15-s + 0.720·16-s + 4.53·17-s − 0.779·18-s + 2.24·19-s + 4.14·20-s + 2.49·21-s − 1.51·22-s + 1.85·23-s − 2.64·24-s + 3.88·25-s + 2.72·26-s − 27-s + 3.46·28-s + ⋯ |
| L(s) = 1 | − 0.551·2-s − 0.577·3-s − 0.695·4-s − 1.33·5-s + 0.318·6-s − 0.942·7-s + 0.935·8-s + 0.333·9-s + 0.735·10-s + 0.586·11-s + 0.401·12-s − 0.968·13-s + 0.519·14-s + 0.769·15-s + 0.180·16-s + 1.10·17-s − 0.183·18-s + 0.515·19-s + 0.927·20-s + 0.543·21-s − 0.323·22-s + 0.386·23-s − 0.539·24-s + 0.776·25-s + 0.534·26-s − 0.192·27-s + 0.655·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4168929552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4168929552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.779T + 2T^{2} \) |
| 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 1.85T + 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 5.12T + 47T^{2} \) |
| 53 | \( 1 - 3.51T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 7.59T + 61T^{2} \) |
| 67 | \( 1 - 0.605T + 67T^{2} \) |
| 71 | \( 1 - 1.80T + 71T^{2} \) |
| 73 | \( 1 + 4.42T + 73T^{2} \) |
| 79 | \( 1 + 0.536T + 79T^{2} \) |
| 83 | \( 1 - 2.30T + 83T^{2} \) |
| 89 | \( 1 + 8.38T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−11.38464565998328628730492291390, −10.23570038055521053666990590829, −9.633775402626684578451249751331, −8.607457758189786414749817981428, −7.59875763136335637351177269531, −6.91534129289711927738717434324, −5.42426411114909420688219141878, −4.33908683931741630421550683143, −3.36519238243805848197391976289, −0.69802288257484510620986445215,
0.69802288257484510620986445215, 3.36519238243805848197391976289, 4.33908683931741630421550683143, 5.42426411114909420688219141878, 6.91534129289711927738717434324, 7.59875763136335637351177269531, 8.607457758189786414749817981428, 9.633775402626684578451249751331, 10.23570038055521053666990590829, 11.38464565998328628730492291390
