| L(s) = 1 | + 2-s + 3·3-s + 4-s + 5-s + 3·6-s + 7-s + 8-s + 6·9-s + 10-s − 2·11-s + 3·12-s + 5·13-s + 14-s + 3·15-s + 16-s + 3·17-s + 6·18-s + 5·19-s + 20-s + 3·21-s − 2·22-s + 2·23-s + 3·24-s + 25-s + 5·26-s + 9·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s + 0.866·12-s + 1.38·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s + 1.41·18-s + 1.14·19-s + 0.223·20-s + 0.654·21-s − 0.426·22-s + 0.417·23-s + 0.612·24-s + 1/5·25-s + 0.980·26-s + 1.73·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.98623458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.98623458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−14.60421113269213, −14.14404758940761, −13.83087454113131, −13.39089176450964, −12.95503274517302, −12.46485955693808, −11.71257531318242, −11.05886894655084, −10.62143726794184, −9.816829327166415, −9.546724910765800, −8.885657322932836, −8.190337672987343, −7.980175419055800, −7.351244126318028, −6.759825890370326, −5.890427648396441, −5.541732460877915, −4.610160443418596, −4.160738905758037, −3.423799224089570, −2.844313794393312, −2.590815627644945, −1.430989672564666, −1.223674065029220,
1.223674065029220, 1.430989672564666, 2.590815627644945, 2.844313794393312, 3.423799224089570, 4.160738905758037, 4.610160443418596, 5.541732460877915, 5.890427648396441, 6.759825890370326, 7.351244126318028, 7.980175419055800, 8.190337672987343, 8.885657322932836, 9.546724910765800, 9.816829327166415, 10.62143726794184, 11.05886894655084, 11.71257531318242, 12.46485955693808, 12.95503274517302, 13.39089176450964, 13.83087454113131, 14.14404758940761, 14.60421113269213
