| L(s) = 1 | + 2.08·2-s − 1.93·3-s + 2.35·4-s + 1.87·5-s − 4.02·6-s − 0.548·7-s + 0.740·8-s + 0.729·9-s + 3.90·10-s − 6.22·11-s − 4.54·12-s − 13-s − 1.14·14-s − 3.61·15-s − 3.16·16-s − 6.48·17-s + 1.52·18-s + 3.43·19-s + 4.40·20-s + 1.05·21-s − 12.9·22-s − 0.591·23-s − 1.42·24-s − 1.49·25-s − 2.08·26-s + 4.38·27-s − 1.29·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 1.11·3-s + 1.17·4-s + 0.836·5-s − 1.64·6-s − 0.207·7-s + 0.261·8-s + 0.243·9-s + 1.23·10-s − 1.87·11-s − 1.31·12-s − 0.277·13-s − 0.306·14-s − 0.933·15-s − 0.791·16-s − 1.57·17-s + 0.358·18-s + 0.788·19-s + 0.985·20-s + 0.231·21-s − 2.77·22-s − 0.123·23-s − 0.291·24-s − 0.299·25-s − 0.409·26-s + 0.843·27-s − 0.244·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 3 | \( 1 + 1.93T + 3T^{2} \) |
| 5 | \( 1 - 1.87T + 5T^{2} \) |
| 7 | \( 1 + 0.548T + 7T^{2} \) |
| 11 | \( 1 + 6.22T + 11T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 + 0.591T + 23T^{2} \) |
| 29 | \( 1 + 3.31T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 + 2.97T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 + 5.16T + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 + 8.76T + 71T^{2} \) |
| 73 | \( 1 + 4.29T + 73T^{2} \) |
| 79 | \( 1 + 1.40T + 79T^{2} \) |
| 83 | \( 1 - 4.60T + 83T^{2} \) |
| 89 | \( 1 + 7.36T + 89T^{2} \) |
| 97 | \( 1 + 4.20T + 97T^{2} \) |
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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
−9.474015898710853318323752651075, −8.278068904431849732372437408810, −7.08483149769833593167993359528, −6.30897273836511237790077214061, −5.63844531420673548203813534243, −5.13811414329394546008770331116, −4.44106090232393123578260960760, −2.99601070462731236282647684872, −2.23214447642374983145443772470, 0,
2.23214447642374983145443772470, 2.99601070462731236282647684872, 4.44106090232393123578260960760, 5.13811414329394546008770331116, 5.63844531420673548203813534243, 6.30897273836511237790077214061, 7.08483149769833593167993359528, 8.278068904431849732372437408810, 9.474015898710853318323752651075
