| L(s) = 1 | − 0.240·2-s − 1.63·3-s − 1.94·4-s − 2.81·5-s + 0.393·6-s − 2.32·7-s + 0.949·8-s − 0.326·9-s + 0.678·10-s − 1.95·11-s + 3.17·12-s + 0.511·13-s + 0.559·14-s + 4.60·15-s + 3.65·16-s − 0.297·17-s + 0.0787·18-s − 2.19·19-s + 5.47·20-s + 3.80·21-s + 0.470·22-s + 1.16·23-s − 1.55·24-s + 2.93·25-s − 0.123·26-s + 5.43·27-s + 4.51·28-s + ⋯ |
| L(s) = 1 | − 0.170·2-s − 0.943·3-s − 0.970·4-s − 1.25·5-s + 0.160·6-s − 0.878·7-s + 0.335·8-s − 0.108·9-s + 0.214·10-s − 0.588·11-s + 0.916·12-s + 0.141·13-s + 0.149·14-s + 1.18·15-s + 0.913·16-s − 0.0720·17-s + 0.0185·18-s − 0.504·19-s + 1.22·20-s + 0.829·21-s + 0.100·22-s + 0.242·23-s − 0.316·24-s + 0.587·25-s − 0.0241·26-s + 1.04·27-s + 0.853·28-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
| \(L(1)\) |
\(\approx\) |
\(0.2579662898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2579662898\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.240T + 2T^{2} \) |
| 3 | \( 1 + 1.63T + 3T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 - 0.511T + 13T^{2} \) |
| 17 | \( 1 + 0.297T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 - 5.46T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 + 5.33T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 - 0.961T + 43T^{2} \) |
| 47 | \( 1 - 0.430T + 47T^{2} \) |
| 53 | \( 1 + 0.576T + 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 0.446T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 3.99T + 73T^{2} \) |
| 79 | \( 1 + 2.31T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 + 0.717T + 89T^{2} \) |
| 97 | \( 1 + 5.15T + 97T^{2} \) |
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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
−10.78403509295271624359950938659, −10.07969416919242711280541443141, −8.929571516032411602125342349337, −8.229308037373693428798596884131, −7.23044182028483787393204562190, −6.14359898976629681732739305126, −5.13364471942729903213221606912, −4.21696833108329651744134724596, −3.17473411483734489680126125767, −0.46825537643274390382496434136,
0.46825537643274390382496434136, 3.17473411483734489680126125767, 4.21696833108329651744134724596, 5.13364471942729903213221606912, 6.14359898976629681732739305126, 7.23044182028483787393204562190, 8.229308037373693428798596884131, 8.929571516032411602125342349337, 10.07969416919242711280541443141, 10.78403509295271624359950938659
