| L(s) = 1 | + 1.26·2-s − 0.320·3-s − 0.404·4-s + 5-s − 0.404·6-s + 7-s − 3.03·8-s − 2.89·9-s + 1.26·10-s − 5.62·11-s + 0.129·12-s − 6.38·13-s + 1.26·14-s − 0.320·15-s − 3.02·16-s − 1.45·17-s − 3.65·18-s + 8.36·19-s − 0.404·20-s − 0.320·21-s − 7.10·22-s − 23-s + 0.973·24-s + 25-s − 8.06·26-s + 1.88·27-s − 0.404·28-s + ⋯ |
| L(s) = 1 | + 0.893·2-s − 0.184·3-s − 0.202·4-s + 0.447·5-s − 0.165·6-s + 0.377·7-s − 1.07·8-s − 0.965·9-s + 0.399·10-s − 1.69·11-s + 0.0374·12-s − 1.77·13-s + 0.337·14-s − 0.0827·15-s − 0.756·16-s − 0.352·17-s − 0.862·18-s + 1.91·19-s − 0.0904·20-s − 0.0699·21-s − 1.51·22-s − 0.208·23-s + 0.198·24-s + 0.200·25-s − 1.58·26-s + 0.363·27-s − 0.0764·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.26T + 2T^{2} \) |
| 3 | \( 1 + 0.320T + 3T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 17 | \( 1 + 1.45T + 17T^{2} \) |
| 19 | \( 1 - 8.36T + 19T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 31 | \( 1 - 0.597T + 31T^{2} \) |
| 37 | \( 1 + 0.962T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.05T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 0.178T + 61T^{2} \) |
| 67 | \( 1 + 9.63T + 67T^{2} \) |
| 71 | \( 1 + 1.23T + 71T^{2} \) |
| 73 | \( 1 + 4.30T + 73T^{2} \) |
| 79 | \( 1 - 0.418T + 79T^{2} \) |
| 83 | \( 1 - 3.61T + 83T^{2} \) |
| 89 | \( 1 - 8.01T + 89T^{2} \) |
| 97 | \( 1 - 8.51T + 97T^{2} \) |
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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
−9.854671440978874401669160382234, −9.096128850933385421722834862587, −8.017464240628278772457039789730, −7.23733328558543487739656386054, −5.82782598428304021332891390373, −5.22853292748432639338332230908, −4.79549537815711170169793759101, −3.16559970076524687777731067940, −2.45101217815759664172286744793, 0,
2.45101217815759664172286744793, 3.16559970076524687777731067940, 4.79549537815711170169793759101, 5.22853292748432639338332230908, 5.82782598428304021332891390373, 7.23733328558543487739656386054, 8.017464240628278772457039789730, 9.096128850933385421722834862587, 9.854671440978874401669160382234
