L(s) = 1 | + 0.575·2-s − 3-s − 1.66·4-s + 2.28·5-s − 0.575·6-s − 4.44·7-s − 2.10·8-s + 9-s + 1.31·10-s + 6.05·11-s + 1.66·12-s − 4.10·13-s − 2.55·14-s − 2.28·15-s + 2.12·16-s + 17-s + 0.575·18-s − 3.28·19-s − 3.82·20-s + 4.44·21-s + 3.48·22-s + 2.60·23-s + 2.10·24-s + 0.241·25-s − 2.36·26-s − 27-s + 7.42·28-s + ⋯ |
L(s) = 1 | + 0.406·2-s − 0.577·3-s − 0.834·4-s + 1.02·5-s − 0.234·6-s − 1.68·7-s − 0.745·8-s + 0.333·9-s + 0.416·10-s + 1.82·11-s + 0.481·12-s − 1.13·13-s − 0.683·14-s − 0.591·15-s + 0.531·16-s + 0.242·17-s + 0.135·18-s − 0.754·19-s − 0.854·20-s + 0.970·21-s + 0.742·22-s + 0.543·23-s + 0.430·24-s + 0.0482·25-s − 0.462·26-s − 0.192·27-s + 1.40·28-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 | \( 1 - 0.575T + 2T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 - 6.05T + 11T^{2} \) |
| 13 | \( 1 + 4.10T + 13T^{2} \) |
| 19 | \( 1 + 3.28T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 4.67T + 29T^{2} \) |
| 31 | \( 1 - 0.308T + 31T^{2} \) |
| 37 | \( 1 + 2.42T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 + 5.80T + 43T^{2} \) |
| 47 | \( 1 - 0.556T + 47T^{2} \) |
| 53 | \( 1 + 1.59T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 7.08T + 61T^{2} \) |
| 67 | \( 1 - 1.94T + 67T^{2} \) |
| 71 | \( 1 - 5.14T + 71T^{2} \) |
| 73 | \( 1 - 3.22T + 73T^{2} \) |
| 79 | \( 1 - 5.44T + 79T^{2} \) |
| 83 | \( 1 + 0.605T + 83T^{2} \) |
| 89 | \( 1 - 2.54T + 89T^{2} \) |
| 97 | \( 1 + 0.0480T + 97T^{2} \) |
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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
−7.04828207138476921465932201085, −6.55250355315729225604051611502, −6.18738462115547761560444785936, −5.43176531684550067105682577413, −4.74273694574405955331736545155, −3.91163243364845493359833408293, −3.31459535057468770996709932203, −2.31997705028111937934925606015, −1.09476122892256593012854920571, 0,
1.09476122892256593012854920571, 2.31997705028111937934925606015, 3.31459535057468770996709932203, 3.91163243364845493359833408293, 4.74273694574405955331736545155, 5.43176531684550067105682577413, 6.18738462115547761560444785936, 6.55250355315729225604051611502, 7.04828207138476921465932201085