| L(s) = 1 | + 3-s − 4·7-s + 9-s + 4·13-s − 8·19-s − 4·21-s + 4·23-s + 27-s − 6·29-s − 8·31-s − 4·37-s + 4·39-s + 6·41-s + 4·43-s − 4·47-s + 9·49-s − 12·53-s − 8·57-s − 6·61-s − 4·63-s + 12·67-s + 4·69-s − 16·71-s − 8·79-s + 81-s − 12·83-s − 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s − 1.83·19-s − 0.872·21-s + 0.834·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.657·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s − 1.64·53-s − 1.05·57-s − 0.768·61-s − 0.503·63-s + 1.46·67-s + 0.481·69-s − 1.89·71-s − 0.900·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.902823429730034903796185580495, −7.84161812555699504527813747179, −6.95756593027595618574514129256, −6.33937261327383596982485750252, −5.63085870340990006632960124727, −4.27460411019146365757900718473, −3.60405203154249968708316901683, −2.84502227093973034349885876665, −1.68377693849669875173825025336, 0,
1.68377693849669875173825025336, 2.84502227093973034349885876665, 3.60405203154249968708316901683, 4.27460411019146365757900718473, 5.63085870340990006632960124727, 6.33937261327383596982485750252, 6.95756593027595618574514129256, 7.84161812555699504527813747179, 8.902823429730034903796185580495
