| L(s) = 1 | + 3·3-s + 3·7-s + 6·9-s − 11-s + 5·13-s + 4·17-s + 8·19-s + 9·21-s + 7·23-s − 5·25-s + 9·27-s − 6·29-s − 2·31-s − 3·33-s − 10·37-s + 15·39-s − 6·41-s − 43-s − 7·47-s + 2·49-s + 12·51-s + 6·53-s + 24·57-s + 4·59-s − 7·61-s + 18·63-s + 7·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.13·7-s + 2·9-s − 0.301·11-s + 1.38·13-s + 0.970·17-s + 1.83·19-s + 1.96·21-s + 1.45·23-s − 25-s + 1.73·27-s − 1.11·29-s − 0.359·31-s − 0.522·33-s − 1.64·37-s + 2.40·39-s − 0.937·41-s − 0.152·43-s − 1.02·47-s + 2/7·49-s + 1.68·51-s + 0.824·53-s + 3.17·57-s + 0.520·59-s − 0.896·61-s + 2.26·63-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.603120898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.603120898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.095771780449032265506272433934, −7.39046541661170756480887890114, −6.79154007364481866657543912802, −5.36343543242665739425666647444, −5.21519885825346971316256007547, −3.85111334322369477101798009441, −3.54062449876866370771791680531, −2.80016845152533898817028379697, −1.66995760821360723640234606556, −1.29639543025074551921365896861,
1.29639543025074551921365896861, 1.66995760821360723640234606556, 2.80016845152533898817028379697, 3.54062449876866370771791680531, 3.85111334322369477101798009441, 5.21519885825346971316256007547, 5.36343543242665739425666647444, 6.79154007364481866657543912802, 7.39046541661170756480887890114, 8.095771780449032265506272433934
