| L(s) = 1 | + 3-s + 7-s + 9-s + 4·11-s + 2·13-s + 6·17-s + 21-s + 27-s − 6·29-s + 4·33-s + 6·37-s + 2·39-s − 10·41-s + 12·47-s + 49-s + 6·51-s + 6·53-s − 4·59-s + 2·61-s + 63-s − 8·67-s − 4·71-s − 10·73-s + 4·77-s + 81-s − 12·83-s − 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.218·21-s + 0.192·27-s − 1.11·29-s + 0.696·33-s + 0.986·37-s + 0.320·39-s − 1.56·41-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.520·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s − 0.474·71-s − 1.17·73-s + 0.455·77-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.037605456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.037605456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.558627258183637603102674436269, −7.60426678792300611346421045582, −7.17905020908804665331447889708, −6.12934038986280881759669496804, −5.55988147328395071637819645056, −4.45525117501421917522650648004, −3.77091079728462215483556362033, −3.06835713647073940497916059788, −1.84815801767940451921502681724, −1.05833149547745674631258729471,
1.05833149547745674631258729471, 1.84815801767940451921502681724, 3.06835713647073940497916059788, 3.77091079728462215483556362033, 4.45525117501421917522650648004, 5.55988147328395071637819645056, 6.12934038986280881759669496804, 7.17905020908804665331447889708, 7.60426678792300611346421045582, 8.558627258183637603102674436269
