| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 9-s − 6·11-s + 2·12-s − 12·13-s − 4·16-s − 2·18-s + 12·22-s + 8·23-s − 9·25-s + 24·26-s + 27-s + 8·32-s − 6·33-s + 2·36-s + 16·37-s − 12·39-s − 12·44-s − 16·46-s + 6·47-s − 4·48-s − 5·49-s + 18·50-s − 24·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.80·11-s + 0.577·12-s − 3.32·13-s − 16-s − 0.471·18-s + 2.55·22-s + 1.66·23-s − 9/5·25-s + 4.70·26-s + 0.192·27-s + 1.41·32-s − 1.04·33-s + 1/3·36-s + 2.63·37-s − 1.92·39-s − 1.80·44-s − 2.35·46-s + 0.875·47-s − 0.577·48-s − 5/7·49-s + 2.54·50-s − 3.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4965118511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4965118511\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.400588261603299107064903686230, −8.892718237653967146997373477022, −8.062798749262723709584757589645, −7.935555959343104225865564799258, −7.42319981131404835915630105083, −7.32016227740693371415400237232, −6.66260307528373962910372616260, −5.74803450606645387288578022188, −5.02993077790447449830764321556, −4.85468716698404797055779902636, −4.12511627512637579479297949689, −2.90182171573806321583393234308, −2.47342224903748840497823197638, −2.13307257724406321246824925475, −0.53708427580882030700657091576,
0.53708427580882030700657091576, 2.13307257724406321246824925475, 2.47342224903748840497823197638, 2.90182171573806321583393234308, 4.12511627512637579479297949689, 4.85468716698404797055779902636, 5.02993077790447449830764321556, 5.74803450606645387288578022188, 6.66260307528373962910372616260, 7.32016227740693371415400237232, 7.42319981131404835915630105083, 7.935555959343104225865564799258, 8.062798749262723709584757589645, 8.892718237653967146997373477022, 9.400588261603299107064903686230
