| L(s) = 1 | − 2·3-s − 2·4-s − 3·9-s − 2·11-s + 4·12-s + 4·16-s − 8·17-s + 4·19-s − 9·25-s + 14·27-s + 4·33-s + 6·36-s + 20·41-s − 8·43-s + 4·44-s − 8·48-s − 10·49-s + 16·51-s − 8·57-s − 2·59-s − 8·64-s − 2·67-s + 16·68-s − 32·73-s + 18·75-s − 8·76-s − 4·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 9-s − 0.603·11-s + 1.15·12-s + 16-s − 1.94·17-s + 0.917·19-s − 9/5·25-s + 2.69·27-s + 0.696·33-s + 36-s + 3.12·41-s − 1.21·43-s + 0.603·44-s − 1.15·48-s − 1.42·49-s + 2.24·51-s − 1.05·57-s − 0.260·59-s − 64-s − 0.244·67-s + 1.94·68-s − 3.74·73-s + 2.07·75-s − 0.917·76-s − 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1308736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1308736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−7.44648329782426661272373183158, −7.21661224338768342466420756206, −6.46838419268852437057061476829, −5.94370912777399366835769176593, −5.82135849339387357739621546932, −5.49927884962180738090473987706, −4.75768041533452306802121891710, −4.58597385457409318027109585808, −4.10120986415973978790495886343, −3.29304594675951575785139026233, −2.84256580616515370878491454559, −2.21612589981477744409569670299, −1.19900710774742687001457822190, 0, 0,
1.19900710774742687001457822190, 2.21612589981477744409569670299, 2.84256580616515370878491454559, 3.29304594675951575785139026233, 4.10120986415973978790495886343, 4.58597385457409318027109585808, 4.75768041533452306802121891710, 5.49927884962180738090473987706, 5.82135849339387357739621546932, 5.94370912777399366835769176593, 6.46838419268852437057061476829, 7.21661224338768342466420756206, 7.44648329782426661272373183158
