| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s − 2·17-s − 2·19-s − 22-s − 24-s − 25-s + 27-s − 33-s + 2·34-s + 2·38-s + 41-s + 43-s + 48-s − 49-s + 50-s + 2·51-s − 54-s + 2·57-s + 59-s + 64-s + 66-s + 67-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s + 11-s − 16-s − 2·17-s − 2·19-s − 22-s − 24-s − 25-s + 27-s − 33-s + 2·34-s + 2·38-s + 41-s + 43-s + 48-s − 49-s + 50-s + 2·51-s − 54-s + 2·57-s + 59-s + 64-s + 66-s + 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1256607866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1256607866\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$ | \( ( 1 - T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−15.34043328614189626901675784065, −14.49356786122127795921995434996, −14.34571041592205361528943903242, −13.25301324941013762299050072315, −13.14488498825266391224630030241, −12.39764694798549826442099294177, −11.46457120346775008809481426109, −11.43543784024726873689769103533, −10.58013683892231668704707658551, −10.42530837244236248432447086564, −9.352510038710622356505889286642, −9.026004063438331298527805431678, −8.479371373944274566855562461374, −7.78292414143922961147563368342, −6.70640051462313961974619854325, −6.52229259023448238669588328170, −5.63538510576822836356840705680, −4.47388984368426759808121563855, −4.17397350823654265270160808926, −2.13460287732343082794920450963,
2.13460287732343082794920450963, 4.17397350823654265270160808926, 4.47388984368426759808121563855, 5.63538510576822836356840705680, 6.52229259023448238669588328170, 6.70640051462313961974619854325, 7.78292414143922961147563368342, 8.479371373944274566855562461374, 9.026004063438331298527805431678, 9.352510038710622356505889286642, 10.42530837244236248432447086564, 10.58013683892231668704707658551, 11.43543784024726873689769103533, 11.46457120346775008809481426109, 12.39764694798549826442099294177, 13.14488498825266391224630030241, 13.25301324941013762299050072315, 14.34571041592205361528943903242, 14.49356786122127795921995434996, 15.34043328614189626901675784065
