| L(s) = 1 | + 1.73·2-s + 0.987·3-s + 1.00·4-s + 2.62·5-s + 1.71·6-s − 1.48·7-s − 1.71·8-s − 2.02·9-s + 4.55·10-s + 4.92·11-s + 0.996·12-s + 4.98·13-s − 2.57·14-s + 2.59·15-s − 4.99·16-s + 0.723·17-s − 3.51·18-s + 2.66·19-s + 2.64·20-s − 1.46·21-s + 8.54·22-s − 6.02·23-s − 1.69·24-s + 1.88·25-s + 8.64·26-s − 4.96·27-s − 1.49·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.570·3-s + 0.504·4-s + 1.17·5-s + 0.699·6-s − 0.560·7-s − 0.607·8-s − 0.675·9-s + 1.43·10-s + 1.48·11-s + 0.287·12-s + 1.38·13-s − 0.687·14-s + 0.668·15-s − 1.24·16-s + 0.175·17-s − 0.828·18-s + 0.611·19-s + 0.592·20-s − 0.319·21-s + 1.82·22-s − 1.25·23-s − 0.346·24-s + 0.376·25-s + 1.69·26-s − 0.954·27-s − 0.282·28-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
| \(L(1)\) |
\(\approx\) |
\(3.371162752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.371162752\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 - 0.987T + 3T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 17 | \( 1 - 0.723T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 23 | \( 1 + 6.02T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 + 7.85T + 41T^{2} \) |
| 43 | \( 1 - 3.94T + 43T^{2} \) |
| 47 | \( 1 + 1.56T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 - 0.167T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 2.15T + 83T^{2} \) |
| 89 | \( 1 + 9.27T + 89T^{2} \) |
| 97 | \( 1 - 2.94T + 97T^{2} \) |
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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
−10.99775924873206117520387558275, −9.653220040541353468735805229768, −9.187226964105729631472462200090, −8.328519108098417012470113748654, −6.58610331228540720824793626058, −6.12557676229561429421547537420, −5.32108417380335725917072646516, −3.82338904645555191840013661061, −3.27510706696125418432909688509, −1.84579483197851444022457351505,
1.84579483197851444022457351505, 3.27510706696125418432909688509, 3.82338904645555191840013661061, 5.32108417380335725917072646516, 6.12557676229561429421547537420, 6.58610331228540720824793626058, 8.328519108098417012470113748654, 9.187226964105729631472462200090, 9.653220040541353468735805229768, 10.99775924873206117520387558275
