L(s) = 1 | − 2·2-s + 2·4-s + 3·5-s − 7-s − 6·10-s + 2·13-s + 2·14-s − 4·16-s − 3·17-s − 4·19-s + 6·20-s − 2·23-s + 4·25-s − 4·26-s − 2·28-s − 8·29-s + 2·31-s + 8·32-s + 6·34-s − 3·35-s + 10·37-s + 8·38-s − 2·41-s + 9·43-s + 4·46-s − 9·47-s + 49-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s + 1.34·5-s − 0.377·7-s − 1.89·10-s + 0.554·13-s + 0.534·14-s − 16-s − 0.727·17-s − 0.917·19-s + 1.34·20-s − 0.417·23-s + 4/5·25-s − 0.784·26-s − 0.377·28-s − 1.48·29-s + 0.359·31-s + 1.41·32-s + 1.02·34-s − 0.507·35-s + 1.64·37-s + 1.29·38-s − 0.312·41-s + 1.37·43-s + 0.589·46-s − 1.31·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.63590327162668529482196407477, −6.94942746030767269213041823496, −6.14935463529028811581800232597, −5.87901729810511972904163345294, −4.70997636001471276033834884131, −3.91961012314561270307060914469, −2.60879152012108403981305861661, −2.03323666012786418834526537363, −1.22575838935660182044456028025, 0,
1.22575838935660182044456028025, 2.03323666012786418834526537363, 2.60879152012108403981305861661, 3.91961012314561270307060914469, 4.70997636001471276033834884131, 5.87901729810511972904163345294, 6.14935463529028811581800232597, 6.94942746030767269213041823496, 7.63590327162668529482196407477