L(s) = 1 | + 2.33·2-s + 3-s + 3.44·4-s − 3.01·5-s + 2.33·6-s + 0.805·7-s + 3.38·8-s + 9-s − 7.03·10-s + 0.210·11-s + 3.44·12-s − 2.50·13-s + 1.88·14-s − 3.01·15-s + 0.998·16-s − 17-s + 2.33·18-s + 1.78·19-s − 10.3·20-s + 0.805·21-s + 0.490·22-s − 1.54·23-s + 3.38·24-s + 4.07·25-s − 5.83·26-s + 27-s + 2.77·28-s + ⋯ |
L(s) = 1 | + 1.65·2-s + 0.577·3-s + 1.72·4-s − 1.34·5-s + 0.952·6-s + 0.304·7-s + 1.19·8-s + 0.333·9-s − 2.22·10-s + 0.0633·11-s + 0.995·12-s − 0.693·13-s + 0.502·14-s − 0.777·15-s + 0.249·16-s − 0.242·17-s + 0.550·18-s + 0.409·19-s − 2.32·20-s + 0.175·21-s + 0.104·22-s − 0.322·23-s + 0.690·24-s + 0.815·25-s − 1.14·26-s + 0.192·27-s + 0.525·28-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 | \( 1 - 2.33T + 2T^{2} \) |
| 5 | \( 1 + 3.01T + 5T^{2} \) |
| 7 | \( 1 - 0.805T + 7T^{2} \) |
| 11 | \( 1 - 0.210T + 11T^{2} \) |
| 13 | \( 1 + 2.50T + 13T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 + 7.38T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 - 5.34T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 1.18T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 8.24T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.32933127689712478977257953612, −6.96114747935257729654595282583, −5.82874359953560371487800642535, −5.21824310097968436182430918168, −4.50191660422159213213537004876, −3.82768337786239208498744693837, −3.50444533633030735551816970823, −2.57663563828552492915998519871, −1.74582309847828606843510266144, 0,
1.74582309847828606843510266144, 2.57663563828552492915998519871, 3.50444533633030735551816970823, 3.82768337786239208498744693837, 4.50191660422159213213537004876, 5.21824310097968436182430918168, 5.82874359953560371487800642535, 6.96114747935257729654595282583, 7.32933127689712478977257953612