| L(s) = 1 | + 4.40·2-s + 3·3-s + 11.4·4-s + 18.5·5-s + 13.2·6-s + 7·7-s + 15.1·8-s + 9·9-s + 81.7·10-s + 19.2·11-s + 34.2·12-s − 61.3·13-s + 30.8·14-s + 55.6·15-s − 24.7·16-s − 7.90·17-s + 39.6·18-s + 89.4·19-s + 212.·20-s + 21·21-s + 84.6·22-s − 23·23-s + 45.3·24-s + 219.·25-s − 270.·26-s + 27·27-s + 80.0·28-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 0.577·3-s + 1.42·4-s + 1.65·5-s + 0.899·6-s + 0.377·7-s + 0.668·8-s + 0.333·9-s + 2.58·10-s + 0.526·11-s + 0.824·12-s − 1.30·13-s + 0.589·14-s + 0.957·15-s − 0.387·16-s − 0.112·17-s + 0.519·18-s + 1.08·19-s + 2.37·20-s + 0.218·21-s + 0.820·22-s − 0.208·23-s + 0.385·24-s + 1.75·25-s − 2.04·26-s + 0.192·27-s + 0.540·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.453814327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.453814327\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 4.40T + 8T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 11 | \( 1 - 19.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.90T + 4.91e3T^{2} \) |
| 19 | \( 1 - 89.4T + 6.85e3T^{2} \) |
| 29 | \( 1 + 70.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 36.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 172.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 103.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 385.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 804.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 227.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 864.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 400.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 24.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 672.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.73e3T + 9.12e5T^{2} \) |
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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
−10.65304364522765516966042831604, −9.589987474321525549465418335846, −9.087234473454177048631282795658, −7.46643966738452841390094333094, −6.61818856669672631094272899620, −5.51688688279104212024588010896, −5.04104388763651112943108122051, −3.73649218511886763155393397646, −2.55984838834625518685820116567, −1.76808192897448668389859660743,
1.76808192897448668389859660743, 2.55984838834625518685820116567, 3.73649218511886763155393397646, 5.04104388763651112943108122051, 5.51688688279104212024588010896, 6.61818856669672631094272899620, 7.46643966738452841390094333094, 9.087234473454177048631282795658, 9.589987474321525549465418335846, 10.65304364522765516966042831604
