L(s) = 1 | + 2-s − 3-s − 4-s − 3·5-s − 6-s + 7-s − 3·8-s − 2·9-s − 3·10-s + 2·11-s + 12-s + 14-s + 3·15-s − 16-s − 2·18-s + 2·19-s + 3·20-s − 21-s + 2·22-s + 3·24-s + 4·25-s + 5·27-s − 28-s + 3·30-s + 4·31-s + 5·32-s − 2·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s − 2/3·9-s − 0.948·10-s + 0.603·11-s + 0.288·12-s + 0.267·14-s + 0.774·15-s − 1/4·16-s − 0.471·18-s + 0.458·19-s + 0.670·20-s − 0.218·21-s + 0.426·22-s + 0.612·24-s + 4/5·25-s + 0.962·27-s − 0.188·28-s + 0.547·30-s + 0.718·31-s + 0.883·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6011 | \( 1+O(T) \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.82500718246752301137281234777, −6.86366033370855986828549159556, −6.19032897301397932437772164000, −5.37876592487891565492869603443, −4.81527691281499224079478591530, −4.09882078680583618247227127786, −3.52171467078518325302187179800, −2.67821425351759730387060966808, −1.00281674235999969089438070007, 0,
1.00281674235999969089438070007, 2.67821425351759730387060966808, 3.52171467078518325302187179800, 4.09882078680583618247227127786, 4.81527691281499224079478591530, 5.37876592487891565492869603443, 6.19032897301397932437772164000, 6.86366033370855986828549159556, 7.82500718246752301137281234777