L(s) = 1 | − 3-s + 4·5-s + 9-s − 4·15-s − 8·19-s + 16·23-s + 2·25-s − 27-s − 12·29-s + 8·43-s + 4·45-s − 14·49-s + 4·53-s + 8·57-s − 8·67-s − 16·69-s − 16·71-s + 20·73-s − 2·75-s + 81-s + 12·87-s − 32·95-s + 4·97-s + 36·101-s + 64·115-s − 6·121-s − 28·125-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.03·15-s − 1.83·19-s + 3.33·23-s + 2/5·25-s − 0.192·27-s − 2.22·29-s + 1.21·43-s + 0.596·45-s − 2·49-s + 0.549·53-s + 1.05·57-s − 0.977·67-s − 1.92·69-s − 1.89·71-s + 2.34·73-s − 0.230·75-s + 1/9·81-s + 1.28·87-s − 3.28·95-s + 0.406·97-s + 3.58·101-s + 5.96·115-s − 0.545·121-s − 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.160141318\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.160141318\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−11.17432181718975973992131540327, −10.68009321996461187722201460512, −10.16546244400809085298891139382, −9.549120050570062839357860330869, −9.065170023749915638598404079220, −8.777512613188475551383845278968, −7.62403207119791885504595168937, −7.13146007277855219299925953915, −6.21040581024747445781537308376, −6.14969218484336320739052245813, −5.24179959934618595033608799542, −4.83326855247570334999188527992, −3.71523857372513194753595622108, −2.51659494204614894722398853037, −1.64229196717434907696557687175,
1.64229196717434907696557687175, 2.51659494204614894722398853037, 3.71523857372513194753595622108, 4.83326855247570334999188527992, 5.24179959934618595033608799542, 6.14969218484336320739052245813, 6.21040581024747445781537308376, 7.13146007277855219299925953915, 7.62403207119791885504595168937, 8.777512613188475551383845278968, 9.065170023749915638598404079220, 9.549120050570062839357860330869, 10.16546244400809085298891139382, 10.68009321996461187722201460512, 11.17432181718975973992131540327