| L(s) = 1 | − 2·2-s + 2·4-s − 8·11-s + 13-s − 4·16-s + 16·22-s + 5·23-s − 25-s − 2·26-s + 8·32-s + 3·37-s − 16·44-s − 10·46-s + 3·47-s − 4·49-s + 2·50-s + 2·52-s − 12·59-s − 8·61-s − 8·64-s − 20·71-s − 15·73-s − 6·74-s − 8·83-s + 10·92-s − 6·94-s − 29·97-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 2.41·11-s + 0.277·13-s − 16-s + 3.41·22-s + 1.04·23-s − 1/5·25-s − 0.392·26-s + 1.41·32-s + 0.493·37-s − 2.41·44-s − 1.47·46-s + 0.437·47-s − 4/7·49-s + 0.282·50-s + 0.277·52-s − 1.56·59-s − 1.02·61-s − 64-s − 2.37·71-s − 1.75·73-s − 0.697·74-s − 0.878·83-s + 1.04·92-s − 0.618·94-s − 2.94·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−9.911548331967170602224432842943, −9.305966686803148643515478529449, −8.750561119787304486337800828282, −8.377479751783081148472503656271, −7.79024244853088464908604149253, −7.45858465828975888040007206656, −7.04435364361933627988688408736, −6.11467159302532349132508416828, −5.61683884203224701232925448687, −4.85465722416402411124453103536, −4.39595721692298053176383356260, −3.06149618077595600935099454614, −2.62603142691236996321171717870, −1.50823898313521478144137224729, 0,
1.50823898313521478144137224729, 2.62603142691236996321171717870, 3.06149618077595600935099454614, 4.39595721692298053176383356260, 4.85465722416402411124453103536, 5.61683884203224701232925448687, 6.11467159302532349132508416828, 7.04435364361933627988688408736, 7.45858465828975888040007206656, 7.79024244853088464908604149253, 8.377479751783081148472503656271, 8.750561119787304486337800828282, 9.305966686803148643515478529449, 9.911548331967170602224432842943
