| L(s) = 1 | − 1.91·2-s − 0.186·3-s + 1.65·4-s − 2.25·5-s + 0.355·6-s + 3.52·7-s + 0.663·8-s − 2.96·9-s + 4.30·10-s + 0.515·11-s − 0.307·12-s − 5.38·13-s − 6.74·14-s + 0.419·15-s − 4.57·16-s − 4.16·17-s + 5.66·18-s + 4.92·19-s − 3.72·20-s − 0.657·21-s − 0.985·22-s − 7.69·23-s − 0.123·24-s + 0.0674·25-s + 10.2·26-s + 1.11·27-s + 5.83·28-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 0.107·3-s + 0.826·4-s − 1.00·5-s + 0.145·6-s + 1.33·7-s + 0.234·8-s − 0.988·9-s + 1.36·10-s + 0.155·11-s − 0.0888·12-s − 1.49·13-s − 1.80·14-s + 0.108·15-s − 1.14·16-s − 1.01·17-s + 1.33·18-s + 1.13·19-s − 0.831·20-s − 0.143·21-s − 0.210·22-s − 1.60·23-s − 0.0252·24-s + 0.0134·25-s + 2.01·26-s + 0.213·27-s + 1.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 3 | \( 1 + 0.186T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 - 0.515T + 11T^{2} \) |
| 13 | \( 1 + 5.38T + 13T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 - 4.92T + 19T^{2} \) |
| 23 | \( 1 + 7.69T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 31 | \( 1 + 4.43T + 31T^{2} \) |
| 37 | \( 1 - 5.99T + 37T^{2} \) |
| 41 | \( 1 + 8.99T + 41T^{2} \) |
| 43 | \( 1 + 1.66T + 43T^{2} \) |
| 47 | \( 1 - 8.55T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 3.91T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−11.63102528619926898386477463925, −10.70599727731048017273416349635, −9.550309840797062119224988995294, −8.614027947305890179654945538483, −7.79783467451283422206483284993, −7.29425027975600980879488973566, −5.38892382368832931887551820150, −4.19330997205883169168635222870, −2.11090369589586914936954379992, 0,
2.11090369589586914936954379992, 4.19330997205883169168635222870, 5.38892382368832931887551820150, 7.29425027975600980879488973566, 7.79783467451283422206483284993, 8.614027947305890179654945538483, 9.550309840797062119224988995294, 10.70599727731048017273416349635, 11.63102528619926898386477463925
