| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s − 3·11-s + 12-s + 13-s + 14-s + 16-s − 2·18-s + 2·19-s + 21-s − 3·22-s − 3·23-s + 24-s − 5·25-s + 26-s − 5·27-s + 28-s + 5·31-s + 32-s − 3·33-s − 2·36-s − 7·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.471·18-s + 0.458·19-s + 0.218·21-s − 0.639·22-s − 0.625·23-s + 0.204·24-s − 25-s + 0.196·26-s − 0.962·27-s + 0.188·28-s + 0.898·31-s + 0.176·32-s − 0.522·33-s − 1/3·36-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.920406587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.920406587\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−12.77123135624210831031203254395, −11.73502655834995143495656212449, −10.86871494678819831017883536020, −9.687293312919998106921740803494, −8.357542398475312637001956914796, −7.63380170955425435565378568582, −6.10743185313712551583625613621, −5.06509234546123458101387314677, −3.60858467418748472558474969025, −2.33000973337707432224237511883,
2.33000973337707432224237511883, 3.60858467418748472558474969025, 5.06509234546123458101387314677, 6.10743185313712551583625613621, 7.63380170955425435565378568582, 8.357542398475312637001956914796, 9.687293312919998106921740803494, 10.86871494678819831017883536020, 11.73502655834995143495656212449, 12.77123135624210831031203254395
