| L(s) = 1 | − 0.807·2-s − 1.50·3-s − 1.34·4-s + 0.199·5-s + 1.21·6-s − 1.00·7-s + 2.70·8-s − 0.721·9-s − 0.161·10-s + 3.66·11-s + 2.03·12-s − 1.86·13-s + 0.813·14-s − 0.301·15-s + 0.513·16-s − 1.18·17-s + 0.582·18-s + 5.40·19-s − 0.269·20-s + 1.52·21-s − 2.95·22-s − 0.413·23-s − 4.08·24-s − 4.96·25-s + 1.50·26-s + 5.61·27-s + 1.35·28-s + ⋯ |
| L(s) = 1 | − 0.570·2-s − 0.871·3-s − 0.673·4-s + 0.0894·5-s + 0.497·6-s − 0.380·7-s + 0.955·8-s − 0.240·9-s − 0.0510·10-s + 1.10·11-s + 0.587·12-s − 0.517·13-s + 0.217·14-s − 0.0779·15-s + 0.128·16-s − 0.287·17-s + 0.137·18-s + 1.24·19-s − 0.0602·20-s + 0.331·21-s − 0.630·22-s − 0.0862·23-s − 0.832·24-s − 0.992·25-s + 0.295·26-s + 1.08·27-s + 0.256·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 997 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 997 | \( 1 + T \) |
| good | 2 | \( 1 + 0.807T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 5 | \( 1 - 0.199T + 5T^{2} \) |
| 7 | \( 1 + 1.00T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 19 | \( 1 - 5.40T + 19T^{2} \) |
| 23 | \( 1 + 0.413T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 + 9.08T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 0.365T + 61T^{2} \) |
| 67 | \( 1 + 2.04T + 67T^{2} \) |
| 71 | \( 1 + 7.85T + 71T^{2} \) |
| 73 | \( 1 + 4.34T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 1.41T + 83T^{2} \) |
| 89 | \( 1 - 5.10T + 89T^{2} \) |
| 97 | \( 1 + 9.16T + 97T^{2} \) |
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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
−9.694708802269304545256853779008, −8.861805008317777429050682665085, −8.020807130300907546238535396978, −6.98124491615477868324791350073, −6.15839182531389837259967784738, −5.21774063760257288746394118741, −4.40647200777163681271172326471, −3.19917204795585995626605982689, −1.36820250937996606429823204646, 0,
1.36820250937996606429823204646, 3.19917204795585995626605982689, 4.40647200777163681271172326471, 5.21774063760257288746394118741, 6.15839182531389837259967784738, 6.98124491615477868324791350073, 8.020807130300907546238535396978, 8.861805008317777429050682665085, 9.694708802269304545256853779008
