| L(s) = 1 | − 3-s − 4-s + 12-s − 2·13-s − 19-s + 27-s + 2·31-s − 37-s + 2·39-s − 4·43-s + 2·52-s + 57-s − 61-s + 64-s − 67-s + 73-s + 76-s + 79-s − 81-s − 2·93-s − 2·97-s + 103-s − 108-s + 109-s + 111-s − 121-s − 2·124-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s + 12-s − 2·13-s − 19-s + 27-s + 2·31-s − 37-s + 2·39-s − 4·43-s + 2·52-s + 57-s − 61-s + 64-s − 67-s + 73-s + 76-s + 79-s − 81-s − 2·93-s − 2·97-s + 103-s − 108-s + 109-s + 111-s − 121-s − 2·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1578602082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1578602082\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−8.653319836220177306224996332100, −8.608269490605023682766841103989, −8.195978817032358971181742351320, −7.996012678339229751737792939323, −7.15570208824648186719053758802, −7.11963165890743106130832323056, −6.56957402630262210893092052228, −6.39153924256112075569056985683, −5.89393948844653817684359059973, −5.35160424173347701119444353122, −5.03827274400745110159466885421, −4.70315071050926586420988770748, −4.63792981114622709091690035182, −4.12803762927395620998288859810, −3.24515126510555246084648002333, −3.23452761038858395365596346527, −2.37363935845580354016517679418, −2.06296727126826652803087855645, −1.24483209349006825236534939382, −0.25968829259013651211178750589,
0.25968829259013651211178750589, 1.24483209349006825236534939382, 2.06296727126826652803087855645, 2.37363935845580354016517679418, 3.23452761038858395365596346527, 3.24515126510555246084648002333, 4.12803762927395620998288859810, 4.63792981114622709091690035182, 4.70315071050926586420988770748, 5.03827274400745110159466885421, 5.35160424173347701119444353122, 5.89393948844653817684359059973, 6.39153924256112075569056985683, 6.56957402630262210893092052228, 7.11963165890743106130832323056, 7.15570208824648186719053758802, 7.996012678339229751737792939323, 8.195978817032358971181742351320, 8.608269490605023682766841103989, 8.653319836220177306224996332100
