| L(s) = 1 | − 2-s − 4-s − 7-s + 3·8-s + 14-s − 16-s − 6·17-s − 6·23-s + 28-s − 6·31-s − 5·32-s + 6·34-s − 4·37-s − 6·41-s + 4·43-s + 6·46-s + 8·47-s + 49-s − 2·53-s − 3·56-s − 2·61-s + 6·62-s + 7·64-s − 4·67-s + 6·68-s − 8·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 1.25·23-s + 0.188·28-s − 1.07·31-s − 0.883·32-s + 1.02·34-s − 0.657·37-s − 0.937·41-s + 0.609·43-s + 0.884·46-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 0.400·56-s − 0.256·61-s + 0.762·62-s + 7/8·64-s − 0.488·67-s + 0.727·68-s − 0.949·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.60942383144785, −13.09595119286761, −12.73788478233511, −12.16564203305927, −11.65748810815309, −11.11414934241426, −10.52448228786338, −10.34036562335797, −9.766624465022236, −9.114473883017327, −8.990021980349179, −8.498754634985098, −7.874333990944431, −7.470061963668353, −6.921768834056855, −6.448830339363292, −5.767083957050276, −5.362991523280506, −4.594647356122391, −4.237560972688751, −3.738733046045656, −3.071822148906534, −2.219269618427134, −1.831298694356524, −1.061255343704684, 0, 0,
1.061255343704684, 1.831298694356524, 2.219269618427134, 3.071822148906534, 3.738733046045656, 4.237560972688751, 4.594647356122391, 5.362991523280506, 5.767083957050276, 6.448830339363292, 6.921768834056855, 7.470061963668353, 7.874333990944431, 8.498754634985098, 8.990021980349179, 9.114473883017327, 9.766624465022236, 10.34036562335797, 10.52448228786338, 11.11414934241426, 11.65748810815309, 12.16564203305927, 12.73788478233511, 13.09595119286761, 13.60942383144785
