| L(s) = 1 | − 4·4-s − 34·13-s + 16·16-s − 46·17-s + 14·25-s + 82·29-s + 48·37-s + 18·41-s + 136·52-s + 146·53-s − 64·64-s + 184·68-s − 81·81-s − 82·89-s − 14·97-s − 56·100-s + 238·101-s + 62·109-s − 448·113-s − 328·116-s + 127-s + 131-s + 137-s + 139-s − 192·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 2.61·13-s + 16-s − 2.70·17-s + 0.559·25-s + 2.82·29-s + 1.29·37-s + 0.439·41-s + 2.61·52-s + 2.75·53-s − 64-s + 2.70·68-s − 81-s − 0.921·89-s − 0.144·97-s − 0.559·100-s + 2.35·101-s + 0.568·109-s − 3.96·113-s − 2.82·116-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.29·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8408898270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8408898270\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 41 | $C_2$ | \( 1 - 18 T + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )( 1 + 8 T + p^{2} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )( 1 + 24 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 - 40 T + p^{2} T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 - 56 T + p^{2} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )( 1 + 96 T + p^{2} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )( 1 + 160 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{2} T^{2} )( 1 + 144 T + p^{2} T^{2} ) \) |
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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
−13.32713503129477579703244545371, −12.42462927162126990760306569705, −11.90669729585967692369474284593, −11.55091056762886278580644403808, −10.60767124873225958658584985250, −10.38250606970917070299852400010, −9.758160270188807408098762334373, −9.333928547069379982267774851008, −8.766036394065137647936941541285, −8.390382229451237419895824250134, −7.70145949474519362473066932866, −6.92945002782437342266292738855, −6.73426613690275383974844495378, −5.75631823665836719408613908189, −4.95503798385721396921366084902, −4.51197656387233899823248179164, −4.26557615877411032759938257378, −2.77188718262874397026813023417, −2.38529024706834496128766867309, −0.56557806712846605293109783590,
0.56557806712846605293109783590, 2.38529024706834496128766867309, 2.77188718262874397026813023417, 4.26557615877411032759938257378, 4.51197656387233899823248179164, 4.95503798385721396921366084902, 5.75631823665836719408613908189, 6.73426613690275383974844495378, 6.92945002782437342266292738855, 7.70145949474519362473066932866, 8.390382229451237419895824250134, 8.766036394065137647936941541285, 9.333928547069379982267774851008, 9.758160270188807408098762334373, 10.38250606970917070299852400010, 10.60767124873225958658584985250, 11.55091056762886278580644403808, 11.90669729585967692369474284593, 12.42462927162126990760306569705, 13.32713503129477579703244545371
