| L(s) = 1 | + 2·3-s + 3·9-s + 2·13-s − 12·17-s + 10·25-s + 4·27-s − 12·29-s + 4·39-s + 8·43-s + 2·49-s − 24·51-s − 12·53-s + 4·61-s + 20·75-s − 16·79-s + 5·81-s − 24·87-s − 12·101-s + 16·103-s − 24·107-s − 12·113-s + 6·117-s + 10·121-s + 127-s + 16·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 0.554·13-s − 2.91·17-s + 2·25-s + 0.769·27-s − 2.22·29-s + 0.640·39-s + 1.21·43-s + 2/7·49-s − 3.36·51-s − 1.64·53-s + 0.512·61-s + 2.30·75-s − 1.80·79-s + 5/9·81-s − 2.57·87-s − 1.19·101-s + 1.57·103-s − 2.32·107-s − 1.12·113-s + 0.554·117-s + 0.909·121-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.849720807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.849720807\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.878723343131379539166875428448, −8.760958007374426773943273257450, −8.685986583575293769474638392246, −7.82147529372751813440576293993, −7.79939317146945905926604743385, −7.20322215129980292762135946322, −6.71991931490483496745692981924, −6.67979676043420737769096360379, −6.18097515676922871963785210228, −5.47869550228222671709826868017, −5.24105676508759016035412698740, −4.44781288339584667435094545926, −4.31937596664120967441103809626, −3.99197194374973620356758048563, −3.31720134533750425500634746113, −2.86897579101799653143808972117, −2.49603684641702680291154491293, −1.87763433979001016339852324693, −1.54519855111347640862829567898, −0.50127494790889295092305902654,
0.50127494790889295092305902654, 1.54519855111347640862829567898, 1.87763433979001016339852324693, 2.49603684641702680291154491293, 2.86897579101799653143808972117, 3.31720134533750425500634746113, 3.99197194374973620356758048563, 4.31937596664120967441103809626, 4.44781288339584667435094545926, 5.24105676508759016035412698740, 5.47869550228222671709826868017, 6.18097515676922871963785210228, 6.67979676043420737769096360379, 6.71991931490483496745692981924, 7.20322215129980292762135946322, 7.79939317146945905926604743385, 7.82147529372751813440576293993, 8.685986583575293769474638392246, 8.760958007374426773943273257450, 8.878723343131379539166875428448
