| L(s) = 1 | − 3·3-s + 2·7-s + 2·9-s − 9·11-s − 2·13-s − 3·17-s − 2·19-s − 6·21-s + 6·23-s − 5·25-s + 6·27-s + 9·29-s − 5·31-s + 27·33-s + 8·37-s + 6·39-s + 3·41-s − 4·43-s + 3·49-s + 9·51-s + 9·53-s + 6·57-s − 12·59-s + 4·63-s − 7·67-s − 18·69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2/3·9-s − 2.71·11-s − 0.554·13-s − 0.727·17-s − 0.458·19-s − 1.30·21-s + 1.25·23-s − 25-s + 1.15·27-s + 1.67·29-s − 0.898·31-s + 4.70·33-s + 1.31·37-s + 0.960·39-s + 0.468·41-s − 0.609·43-s + 3/7·49-s + 1.26·51-s + 1.23·53-s + 0.794·57-s − 1.56·59-s + 0.503·63-s − 0.855·67-s − 2.16·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72454144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72454144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 77 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 45 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 83 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 95 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 77 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 15 T + 191 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 175 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 214 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.56755265955695814080715084321, −7.37801921362988935446133494121, −6.70176600595324313660085665303, −6.62526922921960555720449157396, −6.06168608444178173362698809631, −5.92589956186629036290544299090, −5.35099860852754534778861031689, −5.20102427863295712621416149171, −4.98478907850584612628007443906, −4.76041072362544809455538348443, −4.26233591310308781163153290455, −3.86438929701600513165946476385, −3.06811474690963275231666862952, −2.80184385648531317828012925712, −2.35588189537872691239364843191, −2.19304972068538407109295585648, −1.29610242215682550021092614513, −0.77772248912155229024313289188, 0, 0,
0.77772248912155229024313289188, 1.29610242215682550021092614513, 2.19304972068538407109295585648, 2.35588189537872691239364843191, 2.80184385648531317828012925712, 3.06811474690963275231666862952, 3.86438929701600513165946476385, 4.26233591310308781163153290455, 4.76041072362544809455538348443, 4.98478907850584612628007443906, 5.20102427863295712621416149171, 5.35099860852754534778861031689, 5.92589956186629036290544299090, 6.06168608444178173362698809631, 6.62526922921960555720449157396, 6.70176600595324313660085665303, 7.37801921362988935446133494121, 7.56755265955695814080715084321
