| L(s) = 1 | + 1.99·2-s − 0.531·3-s + 1.99·4-s + 0.113·5-s − 1.06·6-s − 3.12·7-s − 0.00957·8-s − 2.71·9-s + 0.227·10-s + 2.02·11-s − 1.06·12-s + 5.47·13-s − 6.25·14-s − 0.0604·15-s − 4.00·16-s − 4.25·17-s − 5.43·18-s − 1.81·19-s + 0.227·20-s + 1.66·21-s + 4.04·22-s − 1.75·23-s + 0.00508·24-s − 4.98·25-s + 10.9·26-s + 3.03·27-s − 6.24·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.306·3-s + 0.997·4-s + 0.0508·5-s − 0.433·6-s − 1.18·7-s − 0.00338·8-s − 0.905·9-s + 0.0719·10-s + 0.610·11-s − 0.306·12-s + 1.51·13-s − 1.67·14-s − 0.0156·15-s − 1.00·16-s − 1.03·17-s − 1.28·18-s − 0.416·19-s + 0.0507·20-s + 0.362·21-s + 0.862·22-s − 0.365·23-s + 0.00103·24-s − 0.997·25-s + 2.14·26-s + 0.584·27-s − 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1759 | \( 1 + T \) |
| good | 2 | \( 1 - 1.99T + 2T^{2} \) |
| 3 | \( 1 + 0.531T + 3T^{2} \) |
| 5 | \( 1 - 0.113T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 + 4.32T + 29T^{2} \) |
| 31 | \( 1 + 3.99T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 4.14T + 43T^{2} \) |
| 47 | \( 1 + 5.02T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 + 6.53T + 67T^{2} \) |
| 71 | \( 1 - 1.16T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 5.16T + 97T^{2} \) |
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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
−9.005466146516908656650730449311, −8.129619172262496295196296369148, −6.64693813202972202119046682512, −6.30603728893746138703646411572, −5.80255927857673420319978561557, −4.75275692530387377372957102853, −3.70201967475401659036207751157, −3.35455475132659221496475241587, −2.06636640658925882430744945313, 0,
2.06636640658925882430744945313, 3.35455475132659221496475241587, 3.70201967475401659036207751157, 4.75275692530387377372957102853, 5.80255927857673420319978561557, 6.30603728893746138703646411572, 6.64693813202972202119046682512, 8.129619172262496295196296369148, 9.005466146516908656650730449311
