| L(s) = 1 | − 2·3-s + 3·9-s + 4·11-s + 2·13-s + 4·17-s − 8·23-s − 2·25-s − 4·27-s − 4·29-s − 8·31-s − 8·33-s + 4·37-s − 4·39-s + 16·41-s − 8·43-s − 12·47-s − 6·49-s − 8·51-s + 4·53-s − 4·59-s − 4·61-s − 8·67-s + 16·69-s + 4·71-s + 12·73-s + 4·75-s + 5·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.20·11-s + 0.554·13-s + 0.970·17-s − 1.66·23-s − 2/5·25-s − 0.769·27-s − 0.742·29-s − 1.43·31-s − 1.39·33-s + 0.657·37-s − 0.640·39-s + 2.49·41-s − 1.21·43-s − 1.75·47-s − 6/7·49-s − 1.12·51-s + 0.549·53-s − 0.520·59-s − 0.512·61-s − 0.977·67-s + 1.92·69-s + 0.474·71-s + 1.40·73-s + 0.461·75-s + 5/9·81-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
| \(L(1)\) |
\(\approx\) |
\(1.337141641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.337141641\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.132064757955343764718459484399, −8.927087577014521518136321072367, −8.130248255289238952859635806188, −7.972209618680294992054433265291, −7.53239582900727284901224466203, −7.26707826261278552196695869442, −6.60967813022995481805985243685, −6.28917551029572086511283600143, −6.01047093862231415893175612814, −5.83427113815997396496277183649, −5.18153819144461454395856653435, −4.88661667460457854686950327026, −4.29718213479100138473990006404, −3.88042272842888373487727079448, −3.60545430611737042969241687746, −3.13481233720381412174731354071, −2.10756985928360306791363446009, −1.79139507889998586686955738379, −1.19470777361452247015886422385, −0.45849006859332749366239737346,
0.45849006859332749366239737346, 1.19470777361452247015886422385, 1.79139507889998586686955738379, 2.10756985928360306791363446009, 3.13481233720381412174731354071, 3.60545430611737042969241687746, 3.88042272842888373487727079448, 4.29718213479100138473990006404, 4.88661667460457854686950327026, 5.18153819144461454395856653435, 5.83427113815997396496277183649, 6.01047093862231415893175612814, 6.28917551029572086511283600143, 6.60967813022995481805985243685, 7.26707826261278552196695869442, 7.53239582900727284901224466203, 7.972209618680294992054433265291, 8.130248255289238952859635806188, 8.927087577014521518136321072367, 9.132064757955343764718459484399
