| L(s) = 1 | + 3-s − 7-s + 9-s + 6·11-s + 4·13-s − 19-s − 21-s − 2·23-s − 5·25-s + 27-s + 2·29-s + 4·31-s + 6·33-s + 10·37-s + 4·39-s − 6·41-s + 49-s − 6·53-s − 57-s + 4·59-s + 6·61-s − 63-s + 2·67-s − 2·69-s − 8·71-s − 6·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s − 0.229·19-s − 0.218·21-s − 0.417·23-s − 25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 1.04·33-s + 1.64·37-s + 0.640·39-s − 0.937·41-s + 1/7·49-s − 0.824·53-s − 0.132·57-s + 0.520·59-s + 0.768·61-s − 0.125·63-s + 0.244·67-s − 0.240·69-s − 0.949·71-s − 0.702·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658098014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658098014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−8.624225145225054905397562237475, −8.129747626792836624459582094295, −7.12012528572295506112469835189, −6.35039517261092575688402359571, −5.95418403331062040818730635664, −4.51797281136008397058099284800, −3.89449812421189494089375814409, −3.22270173763658668756735728513, −1.99663620570416245068441634467, −1.02508331594045844174701875422,
1.02508331594045844174701875422, 1.99663620570416245068441634467, 3.22270173763658668756735728513, 3.89449812421189494089375814409, 4.51797281136008397058099284800, 5.95418403331062040818730635664, 6.35039517261092575688402359571, 7.12012528572295506112469835189, 8.129747626792836624459582094295, 8.624225145225054905397562237475
