| L(s) = 1 | + 2.15·2-s − 1.51·3-s + 2.65·4-s − 2.23·5-s − 3.25·6-s − 3.60·7-s + 1.41·8-s − 0.717·9-s − 4.82·10-s − 1.50·11-s − 4.01·12-s + 0.00459·13-s − 7.77·14-s + 3.37·15-s − 2.25·16-s + 1.11·17-s − 1.54·18-s + 2.49·19-s − 5.93·20-s + 5.44·21-s − 3.24·22-s + 1.72·23-s − 2.13·24-s − 0.00724·25-s + 0.00991·26-s + 5.61·27-s − 9.57·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s − 0.872·3-s + 1.32·4-s − 0.999·5-s − 1.33·6-s − 1.36·7-s + 0.500·8-s − 0.239·9-s − 1.52·10-s − 0.453·11-s − 1.15·12-s + 0.00127·13-s − 2.07·14-s + 0.871·15-s − 0.564·16-s + 0.271·17-s − 0.365·18-s + 0.573·19-s − 1.32·20-s + 1.18·21-s − 0.691·22-s + 0.359·23-s − 0.436·24-s − 0.00144·25-s + 0.00194·26-s + 1.08·27-s − 1.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 503 | \( 1 + T \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 13 | \( 1 - 0.00459T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 - 2.49T + 19T^{2} \) |
| 23 | \( 1 - 1.72T + 23T^{2} \) |
| 29 | \( 1 + 0.572T + 29T^{2} \) |
| 31 | \( 1 - 3.25T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 + 5.89T + 41T^{2} \) |
| 43 | \( 1 + 7.66T + 43T^{2} \) |
| 47 | \( 1 - 6.44T + 47T^{2} \) |
| 53 | \( 1 + 7.82T + 53T^{2} \) |
| 59 | \( 1 - 0.253T + 59T^{2} \) |
| 61 | \( 1 - 7.08T + 61T^{2} \) |
| 67 | \( 1 - 6.16T + 67T^{2} \) |
| 71 | \( 1 + 9.32T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 1.99T + 79T^{2} \) |
| 83 | \( 1 - 6.91T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−10.88755733215853625200847277800, −9.892496038490913202254136188722, −8.591826703320284408788389356768, −7.25547444473243390696457949041, −6.47242816868127899244487356049, −5.64008773383229354511052733685, −4.81882432559567811103129986471, −3.63041197872983686430071714236, −2.95781778542714237960065650684, 0,
2.95781778542714237960065650684, 3.63041197872983686430071714236, 4.81882432559567811103129986471, 5.64008773383229354511052733685, 6.47242816868127899244487356049, 7.25547444473243390696457949041, 8.591826703320284408788389356768, 9.892496038490913202254136188722, 10.88755733215853625200847277800
