| L(s) = 1 | − 2·3-s − 2·4-s + 3·9-s + 4·12-s + 4·16-s − 25-s − 4·27-s − 6·36-s − 8·48-s − 14·49-s + 24·59-s − 8·64-s − 8·67-s + 2·75-s + 5·81-s − 12·89-s + 20·97-s + 2·100-s + 8·108-s − 24·113-s − 11·121-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 28·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 9-s + 1.15·12-s + 16-s − 1/5·25-s − 0.769·27-s − 36-s − 1.15·48-s − 2·49-s + 3.12·59-s − 64-s − 0.977·67-s + 0.230·75-s + 5/9·81-s − 1.27·89-s + 2.03·97-s + 1/5·100-s + 0.769·108-s − 2.25·113-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 144-s + 2.30·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6913518051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6913518051\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.83325755408735693600071609695, −7.38541850339832998668268026589, −6.85155955042050463476230970468, −6.55439729290212311710146682030, −5.98035480045830507258967618424, −5.62024413796275960251022461755, −5.22535109283974198483551331908, −4.82313147209422087876176340175, −4.38225399232848593248386531452, −3.90280943155393742937099473009, −3.45538269235705626226908879048, −2.76006018578094074868045772964, −1.90633709484756923093390365472, −1.20653669249532522481654849889, −0.42769987983447823604368006289,
0.42769987983447823604368006289, 1.20653669249532522481654849889, 1.90633709484756923093390365472, 2.76006018578094074868045772964, 3.45538269235705626226908879048, 3.90280943155393742937099473009, 4.38225399232848593248386531452, 4.82313147209422087876176340175, 5.22535109283974198483551331908, 5.62024413796275960251022461755, 5.98035480045830507258967618424, 6.55439729290212311710146682030, 6.85155955042050463476230970468, 7.38541850339832998668268026589, 7.83325755408735693600071609695
