| L(s) = 1 | + 2-s + 0.243·3-s + 4-s − 0.634·5-s + 0.243·6-s + 0.795·7-s + 8-s − 2.94·9-s − 0.634·10-s − 4.92·11-s + 0.243·12-s − 6.90·13-s + 0.795·14-s − 0.154·15-s + 16-s − 2.62·17-s − 2.94·18-s + 3.56·19-s − 0.634·20-s + 0.193·21-s − 4.92·22-s − 23-s + 0.243·24-s − 4.59·25-s − 6.90·26-s − 1.44·27-s + 0.795·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.140·3-s + 0.5·4-s − 0.283·5-s + 0.0993·6-s + 0.300·7-s + 0.353·8-s − 0.980·9-s − 0.200·10-s − 1.48·11-s + 0.0702·12-s − 1.91·13-s + 0.212·14-s − 0.0398·15-s + 0.250·16-s − 0.636·17-s − 0.693·18-s + 0.817·19-s − 0.141·20-s + 0.0422·21-s − 1.04·22-s − 0.208·23-s + 0.0496·24-s − 0.919·25-s − 1.35·26-s − 0.278·27-s + 0.150·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.243T + 3T^{2} \) |
| 5 | \( 1 + 0.634T + 5T^{2} \) |
| 7 | \( 1 - 0.795T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 + 6.90T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 2.63T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 3.60T + 53T^{2} \) |
| 59 | \( 1 - 2.41T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 - 3.41T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 8.20T + 83T^{2} \) |
| 89 | \( 1 - 8.62T + 89T^{2} \) |
| 97 | \( 1 + 1.49T + 97T^{2} \) |
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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
−9.304271033766650126261537006435, −8.027369592323741110075744495212, −7.73426520766397712002284909196, −6.77334931793313568775068090791, −5.45751342097612836382367565939, −5.18631090732494188073657267809, −4.09832992105826826671064971682, −2.81948994167351736342867874467, −2.30678736245777323982168759370, 0,
2.30678736245777323982168759370, 2.81948994167351736342867874467, 4.09832992105826826671064971682, 5.18631090732494188073657267809, 5.45751342097612836382367565939, 6.77334931793313568775068090791, 7.73426520766397712002284909196, 8.027369592323741110075744495212, 9.304271033766650126261537006435
