| L(s) = 1 | + 3-s + 4·7-s + 3·9-s + 5·11-s − 10·13-s + 17-s + 6·19-s + 4·21-s − 4·23-s + 5·25-s + 8·27-s + 8·29-s + 5·33-s − 8·37-s − 10·39-s + 8·41-s − 12·43-s + 6·47-s + 9·49-s + 51-s − 11·53-s + 6·57-s − 10·59-s + 12·63-s − 10·67-s − 4·69-s + 18·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 9-s + 1.50·11-s − 2.77·13-s + 0.242·17-s + 1.37·19-s + 0.872·21-s − 0.834·23-s + 25-s + 1.53·27-s + 1.48·29-s + 0.870·33-s − 1.31·37-s − 1.60·39-s + 1.24·41-s − 1.82·43-s + 0.875·47-s + 9/7·49-s + 0.140·51-s − 1.51·53-s + 0.794·57-s − 1.30·59-s + 1.51·63-s − 1.22·67-s − 0.481·69-s + 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.869900054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.869900054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.23531069903192166498734560927, −10.76070078927853563800227546472, −10.16665789367693681511386836437, −9.835149806812365384822016159906, −9.581463170252919500187415214443, −8.920383449795084278142472428340, −8.615433627680208873560939957816, −7.979761100353735473653471487696, −7.43828929365060795735738802250, −7.41990891745679063868411709760, −6.70081315645900508324114976890, −6.34026231463760749894498317388, −5.09884622381532029711602365656, −5.09396971022954596103606947524, −4.57593435215229523296921584938, −4.07215862131710558042867532441, −3.16590876743410920438074084037, −2.61363896895923577898072564749, −1.76410491695041619740276124733, −1.17778144460264655931512169754,
1.17778144460264655931512169754, 1.76410491695041619740276124733, 2.61363896895923577898072564749, 3.16590876743410920438074084037, 4.07215862131710558042867532441, 4.57593435215229523296921584938, 5.09396971022954596103606947524, 5.09884622381532029711602365656, 6.34026231463760749894498317388, 6.70081315645900508324114976890, 7.41990891745679063868411709760, 7.43828929365060795735738802250, 7.979761100353735473653471487696, 8.615433627680208873560939957816, 8.920383449795084278142472428340, 9.581463170252919500187415214443, 9.835149806812365384822016159906, 10.16665789367693681511386836437, 10.76070078927853563800227546472, 11.23531069903192166498734560927
