| L(s) = 1 | − 1.87·3-s + 0.347·7-s + 2.53·9-s + 1.53·11-s + 0.347·17-s − 0.652·21-s − 23-s + 25-s − 2.87·27-s − 1.87·29-s + 1.53·31-s − 2.87·33-s + 1.53·37-s − 41-s − 0.879·49-s − 0.652·51-s + 0.347·59-s − 1.87·61-s + 0.879·63-s + 1.87·69-s − 1.87·75-s + 0.532·77-s + 2.87·81-s + 83-s + 3.53·87-s − 2.87·93-s + 3.87·99-s + ⋯ |
| L(s) = 1 | − 1.87·3-s + 0.347·7-s + 2.53·9-s + 1.53·11-s + 0.347·17-s − 0.652·21-s − 23-s + 25-s − 2.87·27-s − 1.87·29-s + 1.53·31-s − 2.87·33-s + 1.53·37-s − 41-s − 0.879·49-s − 0.652·51-s + 0.347·59-s − 1.87·61-s + 0.879·63-s + 1.87·69-s − 1.87·75-s + 0.532·77-s + 2.87·81-s + 83-s + 3.53·87-s − 2.87·93-s + 3.87·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5206332036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5206332036\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 83 | \( 1 - T \) |
| good | 3 | \( 1 + 1.87T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.347T + T^{2} \) |
| 11 | \( 1 - 1.53T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + 1.87T + T^{2} \) |
| 31 | \( 1 - 1.53T + T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 0.347T + T^{2} \) |
| 61 | \( 1 + 1.87T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−11.71847687385100732727097863544, −11.13524171986824866934843464204, −10.15821515147115438333412651693, −9.288563461584686718668294170469, −7.76252890317528985079731622095, −6.62982332633880580171185982429, −6.04036689619017296177006705362, −4.92234186821467667037822665366, −3.98681698450841188473916072253, −1.37963139780593671644045569350,
1.37963139780593671644045569350, 3.98681698450841188473916072253, 4.92234186821467667037822665366, 6.04036689619017296177006705362, 6.62982332633880580171185982429, 7.76252890317528985079731622095, 9.288563461584686718668294170469, 10.15821515147115438333412651693, 11.13524171986824866934843464204, 11.71847687385100732727097863544
