| L(s) = 1 | − 0.946·2-s − 1.10·4-s + 5-s + 1.56·7-s + 2.93·8-s − 0.946·10-s − 1.50·11-s − 1.47·14-s − 0.571·16-s + 3.26·17-s + 3.10·19-s − 1.10·20-s + 1.42·22-s + 2.93·23-s + 25-s − 1.72·28-s + 2.60·29-s + 2.65·31-s − 5.33·32-s − 3.08·34-s + 1.56·35-s − 0.571·37-s − 2.93·38-s + 2.93·40-s − 5.68·41-s + 1.26·43-s + 1.66·44-s + ⋯ |
| L(s) = 1 | − 0.669·2-s − 0.552·4-s + 0.447·5-s + 0.590·7-s + 1.03·8-s − 0.299·10-s − 0.455·11-s − 0.395·14-s − 0.142·16-s + 0.791·17-s + 0.712·19-s − 0.246·20-s + 0.304·22-s + 0.612·23-s + 0.200·25-s − 0.326·28-s + 0.483·29-s + 0.477·31-s − 0.943·32-s − 0.529·34-s + 0.264·35-s − 0.0939·37-s − 0.476·38-s + 0.464·40-s − 0.888·41-s + 0.192·43-s + 0.251·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.508798624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.508798624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.946T + 2T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 - 2.65T + 31T^{2} \) |
| 37 | \( 1 + 0.571T + 37T^{2} \) |
| 41 | \( 1 + 5.68T + 41T^{2} \) |
| 43 | \( 1 - 1.26T + 43T^{2} \) |
| 47 | \( 1 - 5.13T + 47T^{2} \) |
| 53 | \( 1 - 2.91T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 - 1.22T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 0.796T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−8.132726540157627817389545770047, −7.33657285567867924217588949933, −6.69055666423975587895882114301, −5.50329903993047477695342725042, −5.22680276815153643239546316970, −4.42853369646575107912076730152, −3.52027948953207727785430907280, −2.55778346157665962275163394267, −1.50682038181561099281103116854, −0.74890399771704857465362902993,
0.74890399771704857465362902993, 1.50682038181561099281103116854, 2.55778346157665962275163394267, 3.52027948953207727785430907280, 4.42853369646575107912076730152, 5.22680276815153643239546316970, 5.50329903993047477695342725042, 6.69055666423975587895882114301, 7.33657285567867924217588949933, 8.132726540157627817389545770047
