| L(s) = 1 | − 3·3-s + 4·5-s − 7-s + 6·9-s + 5·13-s − 12·15-s − 5·17-s − 19-s + 3·21-s + 3·23-s + 11·25-s − 9·27-s − 7·29-s + 10·31-s − 4·35-s + 2·37-s − 15·39-s + 6·41-s − 4·43-s + 24·45-s + 8·47-s − 6·49-s + 15·51-s + 9·53-s + 3·57-s + 59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.78·5-s − 0.377·7-s + 2·9-s + 1.38·13-s − 3.09·15-s − 1.21·17-s − 0.229·19-s + 0.654·21-s + 0.625·23-s + 11/5·25-s − 1.73·27-s − 1.29·29-s + 1.79·31-s − 0.676·35-s + 0.328·37-s − 2.40·39-s + 0.937·41-s − 0.609·43-s + 3.57·45-s + 1.16·47-s − 6/7·49-s + 2.10·51-s + 1.23·53-s + 0.397·57-s + 0.130·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.557990271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557990271\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.419476340530386695830013320737, −7.04072677176995484358080095196, −6.53830912314405188123964194618, −6.03500735024241774946796402688, −5.62265885294180576753342648583, −4.83446530624278053777940546615, −4.03331585815584265319099800090, −2.65213461219057874120114990058, −1.64806497467793244246487018893, −0.793840098008458050667769689760,
0.793840098008458050667769689760, 1.64806497467793244246487018893, 2.65213461219057874120114990058, 4.03331585815584265319099800090, 4.83446530624278053777940546615, 5.62265885294180576753342648583, 6.03500735024241774946796402688, 6.53830912314405188123964194618, 7.04072677176995484358080095196, 8.419476340530386695830013320737
