| L(s) = 1 | + 2·3-s + 2·7-s + 9-s − 2·11-s + 2·13-s − 4·19-s + 4·21-s − 6·23-s − 5·25-s − 4·27-s + 8·29-s + 6·31-s − 4·33-s − 8·37-s + 4·39-s + 12·43-s + 8·47-s − 3·49-s + 6·53-s − 8·57-s − 4·59-s + 8·61-s + 2·63-s + 4·67-s − 12·69-s + 6·71-s + 8·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.917·19-s + 0.872·21-s − 1.25·23-s − 25-s − 0.769·27-s + 1.48·29-s + 1.07·31-s − 0.696·33-s − 1.31·37-s + 0.640·39-s + 1.82·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s − 1.05·57-s − 0.520·59-s + 1.02·61-s + 0.251·63-s + 0.488·67-s − 1.44·69-s + 0.712·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.411828194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.411828194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.54451180975496, −15.42104893466144, −14.43782263584098, −14.12786560461554, −13.78549175885471, −13.23733394758262, −12.43330889690656, −12.00748292460886, −11.29952295478431, −10.65364786213522, −10.17694784641871, −9.550142520346992, −8.751350202031268, −8.376049925236838, −8.009003943356416, −7.446321236763978, −6.557077610932288, −5.916275077805136, −5.246461957236030, −4.322178718890290, −3.954121656938223, −3.077587145103627, −2.301299291852922, −1.922100504429779, −0.7077924506077533,
0.7077924506077533, 1.922100504429779, 2.301299291852922, 3.077587145103627, 3.954121656938223, 4.322178718890290, 5.246461957236030, 5.916275077805136, 6.557077610932288, 7.446321236763978, 8.009003943356416, 8.376049925236838, 8.751350202031268, 9.550142520346992, 10.17694784641871, 10.65364786213522, 11.29952295478431, 12.00748292460886, 12.43330889690656, 13.23733394758262, 13.78549175885471, 14.12786560461554, 14.43782263584098, 15.42104893466144, 15.54451180975496
