| L(s) = 1 | − 2·3-s − 2·7-s + 3·9-s − 4·11-s − 4·13-s + 8·17-s + 4·21-s + 4·23-s + 2·25-s − 4·27-s − 4·29-s − 8·31-s + 8·33-s + 4·37-s + 8·39-s + 16·41-s + 16·43-s + 8·47-s + 3·49-s − 16·51-s + 4·53-s + 16·59-s + 4·61-s − 6·63-s − 8·67-s − 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 9-s − 1.20·11-s − 1.10·13-s + 1.94·17-s + 0.872·21-s + 0.834·23-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 + 1.28·39-s + 2.49·41-s + 2.43·43-s + 1.16·47-s + 3/7·49-s − 2.24·51-s + 0.549·53-s + 2.08·59-s + 0.512·61-s − 0.755·63-s − 0.977·67-s − 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170597305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170597305\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.817421068424967559950330760894, −9.572136261086612570640523194149, −9.144481836797608678697651173096, −8.766692608690656435780568592860, −7.86630362255902812121867614464, −7.69456227605940517497832643181, −7.27880137521049875868676758530, −7.19929947333960344906801259896, −6.43016075340021412729815608975, −5.95899517442307074835475374389, −5.49612058227707888847033189946, −5.41840313862141013561020011805, −5.00841810884775736683261922242, −4.19408360835994290506952375745, −3.92076847208155291774300046500, −3.25658274172297822186288434997, −2.45155290542338401064358745570, −2.40806710774772468763678561879, −1.01960223044336731983685784379, −0.59976160867097362359671690645,
0.59976160867097362359671690645, 1.01960223044336731983685784379, 2.40806710774772468763678561879, 2.45155290542338401064358745570, 3.25658274172297822186288434997, 3.92076847208155291774300046500, 4.19408360835994290506952375745, 5.00841810884775736683261922242, 5.41840313862141013561020011805, 5.49612058227707888847033189946, 5.95899517442307074835475374389, 6.43016075340021412729815608975, 7.19929947333960344906801259896, 7.27880137521049875868676758530, 7.69456227605940517497832643181, 7.86630362255902812121867614464, 8.766692608690656435780568592860, 9.144481836797608678697651173096, 9.572136261086612570640523194149, 9.817421068424967559950330760894
