L(s) = 1 | + 2·3-s + 4-s − 8·7-s + 9-s + 2·12-s − 2·13-s + 16-s + 12·19-s − 16·21-s + 25-s − 4·27-s − 8·28-s − 12·31-s + 36-s − 4·37-s − 4·39-s − 20·43-s + 2·48-s + 34·49-s − 2·52-s + 24·57-s + 4·61-s − 8·63-s + 64-s − 24·67-s + 20·73-s + 2·75-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/2·4-s − 3.02·7-s + 1/3·9-s + 0.577·12-s − 0.554·13-s + 1/4·16-s + 2.75·19-s − 3.49·21-s + 1/5·25-s − 0.769·27-s − 1.51·28-s − 2.15·31-s + 1/6·36-s − 0.657·37-s − 0.640·39-s − 3.04·43-s + 0.288·48-s + 34/7·49-s − 0.277·52-s + 3.17·57-s + 0.512·61-s − 1.00·63-s + 1/8·64-s − 2.93·67-s + 2.34·73-s + 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.161137327460865013114506016469, −8.868774471216060607456403615280, −7.83531743824800124245932651769, −7.66670900603824463060194393110, −6.87551315707654393130092819540, −6.87025308528864829107350186696, −6.22340654796584066361422256816, −5.34711726354022861021902847980, −5.33976892050247260934145252130, −3.81800601968201289194081442487, −3.53009310754873249134401547682, −2.96623355538274298367536210534, −2.88253971705917726242541361642, −1.69050510807477611643559790122, 0,
1.69050510807477611643559790122, 2.88253971705917726242541361642, 2.96623355538274298367536210534, 3.53009310754873249134401547682, 3.81800601968201289194081442487, 5.33976892050247260934145252130, 5.34711726354022861021902847980, 6.22340654796584066361422256816, 6.87025308528864829107350186696, 6.87551315707654393130092819540, 7.66670900603824463060194393110, 7.83531743824800124245932651769, 8.868774471216060607456403615280, 9.161137327460865013114506016469