| L(s) = 1 | − 3·3-s + 6·9-s + 6·13-s − 4·17-s − 6·19-s + 3·23-s − 9·27-s − 4·29-s + 9·31-s + 7·37-s − 18·39-s + 2·41-s + 6·43-s + 12·47-s − 7·49-s + 12·51-s + 2·53-s + 18·57-s + 9·59-s + 8·61-s + 15·67-s − 9·69-s + 3·71-s − 6·73-s − 6·79-s + 9·81-s − 6·83-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s + 1.66·13-s − 0.970·17-s − 1.37·19-s + 0.625·23-s − 1.73·27-s − 0.742·29-s + 1.61·31-s + 1.15·37-s − 2.88·39-s + 0.312·41-s + 0.914·43-s + 1.75·47-s − 49-s + 1.68·51-s + 0.274·53-s + 2.38·57-s + 1.17·59-s + 1.02·61-s + 1.83·67-s − 1.08·69-s + 0.356·71-s − 0.702·73-s − 0.675·79-s + 81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.02972291640248, −12.92852024753060, −12.44424254459262, −11.68525957700677, −11.36438446676415, −11.08618640946862, −10.70783168890908, −10.23092867699134, −9.715197473928388, −9.057333325682855, −8.521993588275289, −8.223201430963028, −7.310358744952743, −6.950821844555916, −6.327658028318427, −6.122773056163949, −5.727462391895576, −5.030192158034908, −4.503284994556127, −4.084404466839356, −3.661375950798278, −2.577895165088527, −2.108674687072305, −1.085877199618292, −0.8863572000349708, 0,
0.8863572000349708, 1.085877199618292, 2.108674687072305, 2.577895165088527, 3.661375950798278, 4.084404466839356, 4.503284994556127, 5.030192158034908, 5.727462391895576, 6.122773056163949, 6.327658028318427, 6.950821844555916, 7.310358744952743, 8.223201430963028, 8.521993588275289, 9.057333325682855, 9.715197473928388, 10.23092867699134, 10.70783168890908, 11.08618640946862, 11.36438446676415, 11.68525957700677, 12.44424254459262, 12.92852024753060, 13.02972291640248
