| L(s) = 1 | + 3·9-s + 2·13-s + 6·17-s − 10·25-s − 6·29-s + 20·37-s − 24·41-s − 13·49-s + 18·53-s − 20·61-s + 2·73-s − 8·89-s − 12·97-s − 20·101-s − 18·109-s + 4·113-s + 6·117-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 18·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 9-s + 0.554·13-s + 1.45·17-s − 2·25-s − 1.11·29-s + 3.28·37-s − 3.74·41-s − 1.85·49-s + 2.47·53-s − 2.56·61-s + 0.234·73-s − 0.847·89-s − 1.21·97-s − 1.99·101-s − 1.72·109-s + 0.376·113-s + 0.554·117-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.45·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.74236222044461134007095453218, −7.47536575646542499937872009352, −6.61303503651954034437075836396, −6.60845811420238685670203067170, −5.91824238738469752095279165720, −5.46776906443032883183491997236, −5.22289133265133441977616849928, −4.39011552338323920878309071486, −4.08931083553499496867238827018, −3.63803508541238097399374767207, −3.11827862407422559284554596552, −2.43573806798025905726670590752, −1.52291178817025335850558392010, −1.37942708011457790071691730372, 0,
1.37942708011457790071691730372, 1.52291178817025335850558392010, 2.43573806798025905726670590752, 3.11827862407422559284554596552, 3.63803508541238097399374767207, 4.08931083553499496867238827018, 4.39011552338323920878309071486, 5.22289133265133441977616849928, 5.46776906443032883183491997236, 5.91824238738469752095279165720, 6.60845811420238685670203067170, 6.61303503651954034437075836396, 7.47536575646542499937872009352, 7.74236222044461134007095453218
