| L(s) = 1 | + 0.522·2-s − 3-s − 1.72·4-s − 2.49·5-s − 0.522·6-s + 5.20·7-s − 1.94·8-s + 9-s − 1.30·10-s + 3.79·11-s + 1.72·12-s + 1.98·13-s + 2.71·14-s + 2.49·15-s + 2.43·16-s − 17-s + 0.522·18-s − 4.44·19-s + 4.30·20-s − 5.20·21-s + 1.98·22-s + 1.01·23-s + 1.94·24-s + 1.20·25-s + 1.03·26-s − 27-s − 8.99·28-s + ⋯ |
| L(s) = 1 | + 0.369·2-s − 0.577·3-s − 0.863·4-s − 1.11·5-s − 0.213·6-s + 1.96·7-s − 0.688·8-s + 0.333·9-s − 0.411·10-s + 1.14·11-s + 0.498·12-s + 0.551·13-s + 0.726·14-s + 0.643·15-s + 0.609·16-s − 0.242·17-s + 0.123·18-s − 1.02·19-s + 0.962·20-s − 1.13·21-s + 0.422·22-s + 0.211·23-s + 0.397·24-s + 0.241·25-s + 0.203·26-s − 0.192·27-s − 1.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 0.522T + 2T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 - 5.20T + 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 + 8.84T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 0.328T + 47T^{2} \) |
| 53 | \( 1 - 7.74T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 + 4.50T + 61T^{2} \) |
| 67 | \( 1 - 1.51T + 67T^{2} \) |
| 71 | \( 1 - 0.798T + 71T^{2} \) |
| 73 | \( 1 + 2.42T + 73T^{2} \) |
| 79 | \( 1 + 0.0708T + 79T^{2} \) |
| 83 | \( 1 + 6.45T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 6.02T + 97T^{2} \) |
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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
−7.51965737925393891314696428140, −6.89875223588439371147711237949, −5.79874598571287841815394315434, −5.32103324108462708217385582992, −4.55462224933259444498601036446, −3.93436031686338868749338808295, −3.75684637801509698336824522378, −2.00443860192192522690979072383, −1.17884944952003147370012520732, 0,
1.17884944952003147370012520732, 2.00443860192192522690979072383, 3.75684637801509698336824522378, 3.93436031686338868749338808295, 4.55462224933259444498601036446, 5.32103324108462708217385582992, 5.79874598571287841815394315434, 6.89875223588439371147711237949, 7.51965737925393891314696428140
