L(s) = 1 | + 3-s − 5-s + 9-s − 11-s − 4·13-s − 15-s − 2·17-s − 2·19-s + 4·23-s + 25-s + 27-s − 4·29-s − 8·31-s − 33-s − 10·37-s − 4·39-s − 8·43-s − 45-s − 12·47-s − 2·51-s + 10·53-s + 55-s − 2·57-s − 12·59-s + 2·61-s + 4·65-s − 12·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.43·31-s − 0.174·33-s − 1.64·37-s − 0.640·39-s − 1.21·43-s − 0.149·45-s − 1.75·47-s − 0.280·51-s + 1.37·53-s + 0.134·55-s − 0.264·57-s − 1.56·59-s + 0.256·61-s + 0.496·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.103808455\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.103808455\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.05549508016645, −14.81290287184401, −13.90754134908992, −13.54495440298157, −12.87643683566152, −12.50780406537536, −11.95363613488216, −11.27322463250825, −10.77823324637207, −10.23627688015417, −9.607206292528585, −9.011789642803549, −8.659142792963247, −7.872556708761042, −7.484513438622666, −6.893794871030726, −6.414444913106237, −5.288278256307785, −5.042180059634993, −4.295108404747525, −3.508613944638171, −3.108587606347548, −2.152263010947926, −1.716172209051663, −0.3617594754044390,
0.3617594754044390, 1.716172209051663, 2.152263010947926, 3.108587606347548, 3.508613944638171, 4.295108404747525, 5.042180059634993, 5.288278256307785, 6.414444913106237, 6.893794871030726, 7.484513438622666, 7.872556708761042, 8.659142792963247, 9.011789642803549, 9.607206292528585, 10.23627688015417, 10.77823324637207, 11.27322463250825, 11.95363613488216, 12.50780406537536, 12.87643683566152, 13.54495440298157, 13.90754134908992, 14.81290287184401, 15.05549508016645