| L(s) = 1 | + 2-s + 4-s + 8-s − 4·9-s + 16-s − 8·17-s − 4·18-s − 2·23-s + 25-s − 6·31-s + 32-s − 8·34-s − 4·36-s − 10·41-s − 2·46-s − 10·47-s + 49-s + 50-s − 6·62-s + 64-s − 8·68-s − 2·71-s − 4·72-s − 6·73-s + 6·79-s + 7·81-s − 10·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 4/3·9-s + 1/4·16-s − 1.94·17-s − 0.942·18-s − 0.417·23-s + 1/5·25-s − 1.07·31-s + 0.176·32-s − 1.37·34-s − 2/3·36-s − 1.56·41-s − 0.294·46-s − 1.45·47-s + 1/7·49-s + 0.141·50-s − 0.762·62-s + 1/8·64-s − 0.970·68-s − 0.237·71-s − 0.471·72-s − 0.702·73-s + 0.675·79-s + 7/9·81-s − 1.10·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.815685202010693949004859831017, −8.584209137794959996117619906542, −8.186602608191913161258191032530, −7.41621409910590112505767344554, −6.97296779450865050385441034794, −6.44256293881688599694188315727, −6.02423624747155480664934192423, −5.49405933344192101995519622316, −4.87372938010592245648750249634, −4.49470688963515278068286617452, −3.65255313015543500876200034054, −3.19086208954084191108216954301, −2.40635959264379300468504663340, −1.81531425215631882842922250237, 0,
1.81531425215631882842922250237, 2.40635959264379300468504663340, 3.19086208954084191108216954301, 3.65255313015543500876200034054, 4.49470688963515278068286617452, 4.87372938010592245648750249634, 5.49405933344192101995519622316, 6.02423624747155480664934192423, 6.44256293881688599694188315727, 6.97296779450865050385441034794, 7.41621409910590112505767344554, 8.186602608191913161258191032530, 8.584209137794959996117619906542, 8.815685202010693949004859831017
