| L(s) = 1 | + 3-s − 2·7-s + 9-s + 11-s + 4·13-s − 17-s + 8·19-s − 2·21-s − 6·23-s − 5·25-s + 27-s − 6·29-s + 4·31-s + 33-s − 2·37-s + 4·39-s − 6·41-s − 4·43-s − 6·47-s − 3·49-s − 51-s + 12·53-s + 8·57-s + 4·61-s − 2·63-s − 4·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.242·17-s + 1.83·19-s − 0.436·21-s − 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s − 0.937·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s − 0.140·51-s + 1.64·53-s + 1.05·57-s + 0.512·61-s − 0.251·63-s − 0.488·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.580486837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.580486837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.94233044287670, −14.29845899021923, −13.74872508301613, −13.39251223102360, −13.12871059679849, −12.20979046255103, −11.65499034390133, −11.52485223251416, −10.44788312578735, −10.09181598116489, −9.558472720150105, −9.072589346906938, −8.531115615007349, −7.794347141371580, −7.527933459777796, −6.585915091717297, −6.295073091572246, −5.562176908535987, −4.956997881726312, −3.948114844647468, −3.594313488215427, −3.150972204364572, −2.150404529912987, −1.548181862674317, −0.5791981688184155,
0.5791981688184155, 1.548181862674317, 2.150404529912987, 3.150972204364572, 3.594313488215427, 3.948114844647468, 4.956997881726312, 5.562176908535987, 6.295073091572246, 6.585915091717297, 7.527933459777796, 7.794347141371580, 8.531115615007349, 9.072589346906938, 9.558472720150105, 10.09181598116489, 10.44788312578735, 11.52485223251416, 11.65499034390133, 12.20979046255103, 13.12871059679849, 13.39251223102360, 13.74872508301613, 14.29845899021923, 14.94233044287670
