| L(s) = 1 | − 2.59·2-s + 2.09·3-s + 4.75·4-s − 3.02·5-s − 5.44·6-s − 0.561·7-s − 7.17·8-s + 1.38·9-s + 7.85·10-s + 4.23·11-s + 9.96·12-s − 4.87·13-s + 1.45·14-s − 6.33·15-s + 9.12·16-s − 6.39·17-s − 3.60·18-s + 6.29·19-s − 14.3·20-s − 1.17·21-s − 10.9·22-s − 9.21·23-s − 15.0·24-s + 4.13·25-s + 12.6·26-s − 3.37·27-s − 2.67·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 1.20·3-s + 2.37·4-s − 1.35·5-s − 2.22·6-s − 0.212·7-s − 2.53·8-s + 0.462·9-s + 2.48·10-s + 1.27·11-s + 2.87·12-s − 1.35·13-s + 0.390·14-s − 1.63·15-s + 2.28·16-s − 1.55·17-s − 0.850·18-s + 1.44·19-s − 3.21·20-s − 0.256·21-s − 2.34·22-s − 1.92·23-s − 3.06·24-s + 0.827·25-s + 2.48·26-s − 0.649·27-s − 0.504·28-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 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 + 3.02T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 + 6.39T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 + 9.21T + 23T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 - 2.81T + 41T^{2} \) |
| 43 | \( 1 + 1.57T + 43T^{2} \) |
| 47 | \( 1 + 4.08T + 47T^{2} \) |
| 53 | \( 1 + 4.00T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 0.473T + 61T^{2} \) |
| 67 | \( 1 - 1.80T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 - 0.611T + 73T^{2} \) |
| 79 | \( 1 + 2.49T + 79T^{2} \) |
| 83 | \( 1 - 8.29T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 7.36T + 97T^{2} \) |
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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
−9.796166812587156100633519476369, −9.396780810399117537433914868591, −8.640102991994114300422135320980, −7.75638552375296588600312977709, −7.45687439028094087940947598551, −6.39797918579590970291747243451, −4.20694160453719964937338947382, −3.09905397639925749868674914954, −1.93281725158156473046768149933, 0,
1.93281725158156473046768149933, 3.09905397639925749868674914954, 4.20694160453719964937338947382, 6.39797918579590970291747243451, 7.45687439028094087940947598551, 7.75638552375296588600312977709, 8.640102991994114300422135320980, 9.396780810399117537433914868591, 9.796166812587156100633519476369
