| L(s) = 1 | + 2·3-s − 5-s + 5·7-s + 9-s + 11-s + 4·13-s − 2·15-s − 17-s + 4·19-s + 10·21-s + 6·23-s + 25-s − 4·27-s + 3·29-s + 2·31-s + 2·33-s − 5·35-s − 8·37-s + 8·39-s + 9·41-s + 4·43-s − 45-s − 3·47-s + 18·49-s − 2·51-s + 9·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1.88·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.516·15-s − 0.242·17-s + 0.917·19-s + 2.18·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.557·29-s + 0.359·31-s + 0.348·33-s − 0.845·35-s − 1.31·37-s + 1.28·39-s + 1.40·41-s + 0.609·43-s − 0.149·45-s − 0.437·47-s + 18/7·49-s − 0.280·51-s + 1.23·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.135728312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.135728312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.19832305092900, −13.85016982394709, −13.69504439259909, −12.89712263254931, −12.16367039127240, −11.79759762105281, −11.10307567798773, −10.96661201341526, −10.36747280845332, −9.385752842204405, −8.980936215215567, −8.645187533440844, −8.137963491399749, −7.640687347415045, −7.338823130425810, −6.509384117091241, −5.762448896220070, −5.108275362826550, −4.611284745146396, −3.996251282428757, −3.413136941164659, −2.813178348922845, −2.084144949073571, −1.382219172546701, −0.8724615540505795,
0.8724615540505795, 1.382219172546701, 2.084144949073571, 2.813178348922845, 3.413136941164659, 3.996251282428757, 4.611284745146396, 5.108275362826550, 5.762448896220070, 6.509384117091241, 7.338823130425810, 7.640687347415045, 8.137963491399749, 8.645187533440844, 8.980936215215567, 9.385752842204405, 10.36747280845332, 10.96661201341526, 11.10307567798773, 11.79759762105281, 12.16367039127240, 12.89712263254931, 13.69504439259909, 13.85016982394709, 14.19832305092900
