| L(s) = 1 | + 2·3-s − 2·7-s + 9-s + 6·13-s − 2·17-s + 19-s − 4·21-s + 2·23-s − 4·27-s + 2·29-s + 4·31-s − 10·37-s + 12·39-s − 10·41-s + 6·43-s + 6·47-s − 3·49-s − 4·51-s + 6·53-s + 2·57-s + 4·59-s − 2·61-s − 2·63-s − 2·67-s + 4·69-s + 12·71-s + 6·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.66·13-s − 0.485·17-s + 0.229·19-s − 0.872·21-s + 0.417·23-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 1.64·37-s + 1.92·39-s − 1.56·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.560·51-s + 0.824·53-s + 0.264·57-s + 0.520·59-s − 0.256·61-s − 0.251·63-s − 0.244·67-s + 0.481·69-s + 1.42·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.307442504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.307442504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.26155296621452, −14.49676982801350, −13.87916674319408, −13.67469673816050, −13.19053498981688, −12.66473380740385, −11.91900922423424, −11.43878112254676, −10.65589045165270, −10.32857515805332, −9.561878064525072, −8.978879317945571, −8.711287852217446, −8.180157054649064, −7.561543967362203, −6.702874236154869, −6.473891894756072, −5.646133671742980, −4.998063920619046, −3.996601411099294, −3.593295049014317, −3.090111836267845, −2.372838568660938, −1.602459643126462, −0.6511916577562402,
0.6511916577562402, 1.602459643126462, 2.372838568660938, 3.090111836267845, 3.593295049014317, 3.996601411099294, 4.998063920619046, 5.646133671742980, 6.473891894756072, 6.702874236154869, 7.561543967362203, 8.180157054649064, 8.711287852217446, 8.978879317945571, 9.561878064525072, 10.32857515805332, 10.65589045165270, 11.43878112254676, 11.91900922423424, 12.66473380740385, 13.19053498981688, 13.67469673816050, 13.87916674319408, 14.49676982801350, 15.26155296621452
