L(s) = 1 | − 4·3-s − 2·5-s + 6·9-s + 8·15-s + 10·17-s − 19-s − 7·25-s + 4·27-s + 16·31-s − 12·45-s − 5·49-s − 40·51-s + 4·57-s + 28·59-s − 10·61-s − 12·71-s − 30·73-s + 28·75-s − 8·79-s − 37·81-s − 20·85-s − 64·93-s + 2·95-s − 36·101-s − 28·103-s + 20·107-s − 13·121-s + ⋯ |
L(s) = 1 | − 2.30·3-s − 0.894·5-s + 2·9-s + 2.06·15-s + 2.42·17-s − 0.229·19-s − 7/5·25-s + 0.769·27-s + 2.87·31-s − 1.78·45-s − 5/7·49-s − 5.60·51-s + 0.529·57-s + 3.64·59-s − 1.28·61-s − 1.42·71-s − 3.51·73-s + 3.23·75-s − 0.900·79-s − 4.11·81-s − 2.16·85-s − 6.63·93-s + 0.205·95-s − 3.58·101-s − 2.75·103-s + 1.93·107-s − 1.18·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.148060259363710086433322724847, −7.38601845396983265703380778176, −6.97775492717497319205239152750, −6.58341608756454848978175875806, −5.89152757780490661000300260672, −5.75508495262059909206416744964, −5.51991869338759908788687173901, −4.85172758929064856672456822692, −4.33693159410665846542938642673, −4.05153477392287557754730772390, −3.08915058635532111035563541614, −2.83555716155125599118268588753, −1.44011806260301581515132764673, −0.833864002051176659308197110715, 0,
0.833864002051176659308197110715, 1.44011806260301581515132764673, 2.83555716155125599118268588753, 3.08915058635532111035563541614, 4.05153477392287557754730772390, 4.33693159410665846542938642673, 4.85172758929064856672456822692, 5.51991869338759908788687173901, 5.75508495262059909206416744964, 5.89152757780490661000300260672, 6.58341608756454848978175875806, 6.97775492717497319205239152750, 7.38601845396983265703380778176, 8.148060259363710086433322724847