| L(s) = 1 | − 0.920·2-s − 0.533·3-s − 1.15·4-s + 1.81·5-s + 0.491·6-s + 3.82·7-s + 2.90·8-s − 2.71·9-s − 1.67·10-s + 0.400·11-s + 0.614·12-s − 1.49·13-s − 3.52·14-s − 0.967·15-s − 0.369·16-s − 2.58·17-s + 2.50·18-s + 2.19·19-s − 2.08·20-s − 2.04·21-s − 0.369·22-s + 4.15·23-s − 1.54·24-s − 1.71·25-s + 1.37·26-s + 3.04·27-s − 4.40·28-s + ⋯ |
| L(s) = 1 | − 0.651·2-s − 0.307·3-s − 0.575·4-s + 0.811·5-s + 0.200·6-s + 1.44·7-s + 1.02·8-s − 0.905·9-s − 0.528·10-s + 0.120·11-s + 0.177·12-s − 0.413·13-s − 0.942·14-s − 0.249·15-s − 0.0924·16-s − 0.627·17-s + 0.589·18-s + 0.503·19-s − 0.467·20-s − 0.445·21-s − 0.0786·22-s + 0.865·23-s − 0.316·24-s − 0.342·25-s + 0.269·26-s + 0.586·27-s − 0.833·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.017844955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017844955\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 - T \) |
| good | 2 | \( 1 + 0.920T + 2T^{2} \) |
| 3 | \( 1 + 0.533T + 3T^{2} \) |
| 5 | \( 1 - 1.81T + 5T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 - 0.400T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 - 4.38T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 + 1.71T + 47T^{2} \) |
| 53 | \( 1 + 0.838T + 53T^{2} \) |
| 59 | \( 1 + 0.904T + 59T^{2} \) |
| 61 | \( 1 + 2.03T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 9.78T + 79T^{2} \) |
| 83 | \( 1 + 6.20T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−10.79360889330847309042890234503, −9.822678190508552719679899269740, −9.001762672585468100211937017425, −8.310985042898854439481022036519, −7.47898524122080433454635244986, −6.11288135078791588777069978729, −5.12557795344636042801337176606, −4.47394556894418965711294348286, −2.50993076265656179240401949947, −1.09343665711385546409650069041,
1.09343665711385546409650069041, 2.50993076265656179240401949947, 4.47394556894418965711294348286, 5.12557795344636042801337176606, 6.11288135078791588777069978729, 7.47898524122080433454635244986, 8.310985042898854439481022036519, 9.001762672585468100211937017425, 9.822678190508552719679899269740, 10.79360889330847309042890234503
