L(s) = 1 | + 2·3-s − 3·9-s − 4·13-s + 12·17-s − 6·23-s − 25-s − 14·27-s − 8·39-s − 20·43-s − 10·49-s + 24·51-s − 12·53-s − 8·61-s − 12·69-s − 2·75-s + 4·79-s − 4·81-s + 36·101-s + 16·103-s + 12·107-s − 30·113-s + 12·117-s + 121-s + 127-s − 40·129-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 9-s − 1.10·13-s + 2.91·17-s − 1.25·23-s − 1/5·25-s − 2.69·27-s − 1.28·39-s − 3.04·43-s − 1.42·49-s + 3.36·51-s − 1.64·53-s − 1.02·61-s − 1.44·69-s − 0.230·75-s + 0.450·79-s − 4/9·81-s + 3.58·101-s + 1.57·103-s + 1.16·107-s − 2.82·113-s + 1.10·117-s + 1/11·121-s + 0.0887·127-s − 3.52·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.314187785972322305192087673902, −8.132665807481574488057539990847, −7.66471435961055676315007320566, −7.60852684015633966980596452192, −6.63716263814174422745171385492, −6.11181518853748875342915860937, −5.69510294756229368768383854980, −5.11227132195889762323685903220, −4.77814882467244849114054024293, −3.68139626518860523217158413051, −3.27381985666804460223319703873, −3.10915852533965624890419491672, −2.22738900554088144481773298380, −1.58763508602526038578941432974, 0,
1.58763508602526038578941432974, 2.22738900554088144481773298380, 3.10915852533965624890419491672, 3.27381985666804460223319703873, 3.68139626518860523217158413051, 4.77814882467244849114054024293, 5.11227132195889762323685903220, 5.69510294756229368768383854980, 6.11181518853748875342915860937, 6.63716263814174422745171385492, 7.60852684015633966980596452192, 7.66471435961055676315007320566, 8.132665807481574488057539990847, 8.314187785972322305192087673902