| L(s) = 1 | − 1.35·2-s + 0.387·3-s − 0.157·4-s − 1.46·5-s − 0.526·6-s + 0.194·7-s + 2.92·8-s − 2.84·9-s + 1.99·10-s + 1.87·11-s − 0.0612·12-s + 6.15·13-s − 0.264·14-s − 0.569·15-s − 3.65·16-s − 5.37·17-s + 3.86·18-s − 3.78·19-s + 0.231·20-s + 0.0755·21-s − 2.54·22-s − 3.43·23-s + 1.13·24-s − 2.84·25-s − 8.35·26-s − 2.26·27-s − 0.0307·28-s + ⋯ |
| L(s) = 1 | − 0.959·2-s + 0.223·3-s − 0.0789·4-s − 0.657·5-s − 0.214·6-s + 0.0735·7-s + 1.03·8-s − 0.949·9-s + 0.630·10-s + 0.564·11-s − 0.0176·12-s + 1.70·13-s − 0.0706·14-s − 0.147·15-s − 0.914·16-s − 1.30·17-s + 0.911·18-s − 0.867·19-s + 0.0518·20-s + 0.0164·21-s − 0.541·22-s − 0.715·23-s + 0.231·24-s − 0.568·25-s − 1.63·26-s − 0.436·27-s − 0.00580·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 - 0.387T + 3T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 - 0.194T + 7T^{2} \) |
| 11 | \( 1 - 1.87T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 0.0833T + 29T^{2} \) |
| 31 | \( 1 + 3.95T + 31T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 2.32T + 43T^{2} \) |
| 47 | \( 1 + 8.88T + 47T^{2} \) |
| 53 | \( 1 - 7.04T + 53T^{2} \) |
| 59 | \( 1 - 7.89T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 0.418T + 73T^{2} \) |
| 79 | \( 1 - 0.962T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−10.31163870373631063687635254628, −9.191196666858178373818624449312, −8.468809337844759310198201518207, −8.231946025707911631362248362465, −6.90727053765502447118163051254, −5.92815962031766334523361713221, −4.41316711357575577296065187600, −3.59325802371127207780347488205, −1.79557396086654244112707139958, 0,
1.79557396086654244112707139958, 3.59325802371127207780347488205, 4.41316711357575577296065187600, 5.92815962031766334523361713221, 6.90727053765502447118163051254, 8.231946025707911631362248362465, 8.468809337844759310198201518207, 9.191196666858178373818624449312, 10.31163870373631063687635254628
