| L(s) = 1 | − 1.52·2-s − 2.12·3-s + 0.312·4-s + 0.589·5-s + 3.23·6-s + 2.56·8-s + 1.52·9-s − 0.896·10-s + 1.52·11-s − 0.664·12-s − 1.25·15-s − 4.52·16-s + 4.79·17-s − 2.31·18-s + 1.68·19-s + 0.184·20-s − 2.31·22-s + 1.77·23-s − 5.45·24-s − 4.65·25-s + 3.14·27-s + 6.89·29-s + 1.90·30-s − 6.08·31-s + 1.75·32-s − 3.23·33-s − 7.29·34-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 1.22·3-s + 0.156·4-s + 0.263·5-s + 1.32·6-s + 0.907·8-s + 0.506·9-s − 0.283·10-s + 0.458·11-s − 0.191·12-s − 0.323·15-s − 1.13·16-s + 1.16·17-s − 0.545·18-s + 0.386·19-s + 0.0412·20-s − 0.493·22-s + 0.369·23-s − 1.11·24-s − 0.930·25-s + 0.605·27-s + 1.27·29-s + 0.347·30-s − 1.09·31-s + 0.310·32-s − 0.562·33-s − 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 3 | \( 1 + 2.12T + 3T^{2} \) |
| 5 | \( 1 - 0.589T + 5T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 17 | \( 1 - 4.79T + 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 6.08T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 1.35T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 0.464T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 2.48T + 61T^{2} \) |
| 67 | \( 1 + 7.57T + 67T^{2} \) |
| 71 | \( 1 - 6.60T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 0.973T + 97T^{2} \) |
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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
−7.51116789650292935273687331545, −6.83251816991747628302390376988, −6.12850128810339044300917925846, −5.43938099489233387725037613235, −4.87547424252547275722417765539, −4.01969848250779287701935826843, −2.99914298368258158543816200282, −1.67159926235190607789738035251, −1.00658413089117427348390392908, 0,
1.00658413089117427348390392908, 1.67159926235190607789738035251, 2.99914298368258158543816200282, 4.01969848250779287701935826843, 4.87547424252547275722417765539, 5.43938099489233387725037613235, 6.12850128810339044300917925846, 6.83251816991747628302390376988, 7.51116789650292935273687331545
