L(s) = 1 | − 2.11·2-s − 3-s + 2.48·4-s + 2.70·5-s + 2.11·6-s + 2.58·7-s − 1.01·8-s + 9-s − 5.72·10-s − 4.73·11-s − 2.48·12-s + 0.962·13-s − 5.47·14-s − 2.70·15-s − 2.80·16-s + 17-s − 2.11·18-s − 1.82·19-s + 6.70·20-s − 2.58·21-s + 10.0·22-s − 1.57·23-s + 1.01·24-s + 2.30·25-s − 2.03·26-s − 27-s + 6.41·28-s + ⋯ |
L(s) = 1 | − 1.49·2-s − 0.577·3-s + 1.24·4-s + 1.20·5-s + 0.864·6-s + 0.977·7-s − 0.359·8-s + 0.333·9-s − 1.80·10-s − 1.42·11-s − 0.716·12-s + 0.266·13-s − 1.46·14-s − 0.697·15-s − 0.701·16-s + 0.242·17-s − 0.498·18-s − 0.419·19-s + 1.49·20-s − 0.564·21-s + 2.13·22-s − 0.328·23-s + 0.207·24-s + 0.460·25-s − 0.399·26-s − 0.192·27-s + 1.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 0.962T + 13T^{2} \) |
| 19 | \( 1 + 1.82T + 19T^{2} \) |
| 23 | \( 1 + 1.57T + 23T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 7.71T + 31T^{2} \) |
| 37 | \( 1 + 4.06T + 37T^{2} \) |
| 41 | \( 1 - 2.50T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 - 9.84T + 47T^{2} \) |
| 53 | \( 1 + 2.01T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 1.38T + 71T^{2} \) |
| 73 | \( 1 - 6.22T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 0.517T + 83T^{2} \) |
| 89 | \( 1 + 1.13T + 89T^{2} \) |
| 97 | \( 1 + 5.39T + 97T^{2} \) |
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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
−7.59123384386308307155293424395, −7.09193696854521543438976698332, −6.10955124137405273279343069384, −5.49674951705281301580068391024, −5.01588602290538004636652045672, −3.95856299495892766718994287583, −2.45450941625223709815860250907, −1.98423884944989440209421138090, −1.17315480677342175449357608913, 0,
1.17315480677342175449357608913, 1.98423884944989440209421138090, 2.45450941625223709815860250907, 3.95856299495892766718994287583, 5.01588602290538004636652045672, 5.49674951705281301580068391024, 6.10955124137405273279343069384, 7.09193696854521543438976698332, 7.59123384386308307155293424395