| L(s) = 1 | − 3-s + 7-s + 9-s + 2·11-s − 2·13-s − 6·19-s − 21-s − 27-s − 6·29-s + 10·31-s − 2·33-s + 2·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s − 6·53-s + 6·57-s − 6·61-s + 63-s + 4·67-s + 6·71-s − 14·73-s + 2·77-s + 4·79-s + 81-s + 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.37·19-s − 0.218·21-s − 0.192·27-s − 1.11·29-s + 1.79·31-s − 0.348·33-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.794·57-s − 0.768·61-s + 0.125·63-s + 0.488·67-s + 0.712·71-s − 1.63·73-s + 0.227·77-s + 0.450·79-s + 1/9·81-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.066972018398009382545409531527, −7.23625883041751627947701933234, −6.46406809871360528459208750774, −5.97846627400020268994571865083, −4.88472199467215877321611461730, −4.47461775871016923541940664416, −3.48606836449520796464757548603, −2.32220817433425281855736791143, −1.38085273658773050334628712304, 0,
1.38085273658773050334628712304, 2.32220817433425281855736791143, 3.48606836449520796464757548603, 4.47461775871016923541940664416, 4.88472199467215877321611461730, 5.97846627400020268994571865083, 6.46406809871360528459208750774, 7.23625883041751627947701933234, 8.066972018398009382545409531527
