| L(s) = 1 | + 1.04·2-s + 0.462·3-s − 0.904·4-s + 2.89·5-s + 0.483·6-s − 3.03·7-s − 3.04·8-s − 2.78·9-s + 3.02·10-s − 11-s − 0.417·12-s − 1.99·13-s − 3.17·14-s + 1.33·15-s − 1.37·16-s + 5.93·17-s − 2.91·18-s − 2.38·19-s − 2.61·20-s − 1.40·21-s − 1.04·22-s − 7.16·23-s − 1.40·24-s + 3.35·25-s − 2.08·26-s − 2.67·27-s + 2.74·28-s + ⋯ |
| L(s) = 1 | + 0.740·2-s + 0.266·3-s − 0.452·4-s + 1.29·5-s + 0.197·6-s − 1.14·7-s − 1.07·8-s − 0.928·9-s + 0.956·10-s − 0.301·11-s − 0.120·12-s − 0.552·13-s − 0.849·14-s + 0.344·15-s − 0.343·16-s + 1.43·17-s − 0.687·18-s − 0.546·19-s − 0.584·20-s − 0.306·21-s − 0.223·22-s − 1.49·23-s − 0.286·24-s + 0.671·25-s − 0.408·26-s − 0.514·27-s + 0.518·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 1.04T + 2T^{2} \) |
| 3 | \( 1 - 0.462T + 3T^{2} \) |
| 5 | \( 1 - 2.89T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 13 | \( 1 + 1.99T + 13T^{2} \) |
| 17 | \( 1 - 5.93T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 + 7.16T + 23T^{2} \) |
| 29 | \( 1 + 8.87T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 9.19T + 37T^{2} \) |
| 41 | \( 1 - 5.78T + 41T^{2} \) |
| 43 | \( 1 - 2.81T + 43T^{2} \) |
| 47 | \( 1 + 3.62T + 47T^{2} \) |
| 53 | \( 1 - 4.84T + 53T^{2} \) |
| 59 | \( 1 + 0.138T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 - 3.18T + 67T^{2} \) |
| 71 | \( 1 + 2.24T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 1.84T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−9.324013310083244927018932640989, −8.503518161811214675702248574177, −7.46455897926031713641658704157, −6.12063113420542785219795978754, −5.87327662195537422178894908769, −5.15124878510539677199266308689, −3.81034524270139609728575505779, −3.07219858497438991401745240941, −2.13144869798300243633005699703, 0,
2.13144869798300243633005699703, 3.07219858497438991401745240941, 3.81034524270139609728575505779, 5.15124878510539677199266308689, 5.87327662195537422178894908769, 6.12063113420542785219795978754, 7.46455897926031713641658704157, 8.503518161811214675702248574177, 9.324013310083244927018932640989
