L(s) = 1 | − 2·13-s + 17-s − 8·19-s − 9·23-s − 2·29-s − 9·31-s + 8·37-s − 10·41-s − 6·43-s − 7·47-s − 7·49-s + 3·53-s + 6·59-s + 3·61-s − 4·67-s + 12·71-s + 12·73-s − 11·79-s − 16·83-s − 4·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.554·13-s + 0.242·17-s − 1.83·19-s − 1.87·23-s − 0.371·29-s − 1.61·31-s + 1.31·37-s − 1.56·41-s − 0.914·43-s − 1.02·47-s − 49-s + 0.412·53-s + 0.781·59-s + 0.384·61-s − 0.488·67-s + 1.42·71-s + 1.40·73-s − 1.23·79-s − 1.75·83-s − 0.423·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.25260493567138, −13.65034948457492, −13.06129842583638, −12.71149160529074, −12.34868628462765, −11.57613547805752, −11.36934408287090, −10.73265721959862, −10.11568637118950, −9.841769584967315, −9.356232596852670, −8.470404478527781, −8.346709530199775, −7.784167271367389, −7.094509468714193, −6.630193688719895, −6.124001336589955, −5.554494540687742, −5.000134987270919, −4.349335591354605, −3.856396968012736, −3.354256237749369, −2.406122727710364, −2.030105558649540, −1.402307104211737, 0, 0,
1.402307104211737, 2.030105558649540, 2.406122727710364, 3.354256237749369, 3.856396968012736, 4.349335591354605, 5.000134987270919, 5.554494540687742, 6.124001336589955, 6.630193688719895, 7.094509468714193, 7.784167271367389, 8.346709530199775, 8.470404478527781, 9.356232596852670, 9.841769584967315, 10.11568637118950, 10.73265721959862, 11.36934408287090, 11.57613547805752, 12.34868628462765, 12.71149160529074, 13.06129842583638, 13.65034948457492, 14.25260493567138