| L(s) = 1 | + 2.61·2-s − 3-s + 4.85·4-s − 0.618·5-s − 2.61·6-s + 7-s + 7.47·8-s + 9-s − 1.61·10-s − 4.85·12-s − 0.236·13-s + 2.61·14-s + 0.618·15-s + 9.85·16-s + 1.14·17-s + 2.61·18-s − 5.85·19-s − 3.00·20-s − 21-s + 0.236·23-s − 7.47·24-s − 4.61·25-s − 0.618·26-s − 27-s + 4.85·28-s + 6·29-s + 1.61·30-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.577·3-s + 2.42·4-s − 0.276·5-s − 1.06·6-s + 0.377·7-s + 2.64·8-s + 0.333·9-s − 0.511·10-s − 1.40·12-s − 0.0654·13-s + 0.699·14-s + 0.159·15-s + 2.46·16-s + 0.277·17-s + 0.617·18-s − 1.34·19-s − 0.670·20-s − 0.218·21-s + 0.0492·23-s − 1.52·24-s − 0.923·25-s − 0.121·26-s − 0.192·27-s + 0.917·28-s + 1.11·29-s + 0.295·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.180091990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.180091990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 - 0.236T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 + 6.23T + 37T^{2} \) |
| 41 | \( 1 + 0.236T + 41T^{2} \) |
| 43 | \( 1 - 6.70T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 0.381T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 5.70T + 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 - 7.85T + 97T^{2} \) |
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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
−11.58919886972212187961866128887, −11.05033757182290375695842507180, −10.07821864330822044545877542179, −8.315313212703412726921943271422, −7.16132082040376887648743902098, −6.32134835294888550388313327552, −5.38132149526540359444223064265, −4.51559678261015459462304969752, −3.58051138352353501631455446828, −2.04984432008032180258301225430,
2.04984432008032180258301225430, 3.58051138352353501631455446828, 4.51559678261015459462304969752, 5.38132149526540359444223064265, 6.32134835294888550388313327552, 7.16132082040376887648743902098, 8.315313212703412726921943271422, 10.07821864330822044545877542179, 11.05033757182290375695842507180, 11.58919886972212187961866128887
