| L(s) = 1 | − 0.755·2-s − 2.82·3-s − 1.42·4-s + 5-s + 2.13·6-s + 0.0498·7-s + 2.59·8-s + 4.98·9-s − 0.755·10-s + 4.45·11-s + 4.03·12-s − 2.43·13-s − 0.0376·14-s − 2.82·15-s + 0.898·16-s − 2.48·17-s − 3.77·18-s − 5.16·19-s − 1.42·20-s − 0.140·21-s − 3.36·22-s − 4.99·23-s − 7.32·24-s + 25-s + 1.83·26-s − 5.62·27-s − 0.0711·28-s + ⋯ |
| L(s) = 1 | − 0.534·2-s − 1.63·3-s − 0.714·4-s + 0.447·5-s + 0.872·6-s + 0.0188·7-s + 0.916·8-s + 1.66·9-s − 0.239·10-s + 1.34·11-s + 1.16·12-s − 0.674·13-s − 0.0100·14-s − 0.729·15-s + 0.224·16-s − 0.601·17-s − 0.888·18-s − 1.18·19-s − 0.319·20-s − 0.0307·21-s − 0.717·22-s − 1.04·23-s − 1.49·24-s + 0.200·25-s + 0.360·26-s − 1.08·27-s − 0.0134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 + 0.755T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 - 0.0498T + 7T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 - 2.59T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 + 0.543T + 43T^{2} \) |
| 47 | \( 1 + 9.63T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 + 5.36T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
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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
−10.71554038089888899654972827821, −9.745372462774772993492548212405, −9.178693410589626502589573472935, −7.912661095778710120046847219242, −6.65816916178860077736678377981, −6.04179844967899344415925700148, −4.85190281950692521821296613566, −4.16220540712002498661950244258, −1.59911430237976937126415057322, 0,
1.59911430237976937126415057322, 4.16220540712002498661950244258, 4.85190281950692521821296613566, 6.04179844967899344415925700148, 6.65816916178860077736678377981, 7.912661095778710120046847219242, 9.178693410589626502589573472935, 9.745372462774772993492548212405, 10.71554038089888899654972827821
