| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 2·6-s − 4·8-s + 9-s + 3·12-s + 5·16-s − 2·18-s − 19-s − 4·24-s + 3·25-s + 27-s − 5·29-s − 6·32-s + 3·36-s + 2·38-s + 7·41-s − 43-s + 5·48-s − 5·49-s − 6·50-s − 25·53-s − 2·54-s − 57-s + 10·58-s + 3·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.816·6-s − 1.41·8-s + 1/3·9-s + 0.866·12-s + 5/4·16-s − 0.471·18-s − 0.229·19-s − 0.816·24-s + 3/5·25-s + 0.192·27-s − 0.928·29-s − 1.06·32-s + 1/2·36-s + 0.324·38-s + 1.09·41-s − 0.152·43-s + 0.721·48-s − 5/7·49-s − 0.848·50-s − 3.43·53-s − 0.272·54-s − 0.132·57-s + 1.31·58-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 139 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
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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
−8.230247022199406506458314543070, −7.70448810423057161566656504945, −7.39431735259013957516833938172, −6.88079171131385486535826064688, −6.45721627168504731854771604291, −6.03267634618817585971741043189, −5.42726382874489677927526159971, −4.83597441096562558631379203845, −4.22195688919300391079583105271, −3.56825199163943836512869689869, −3.01619245724457508796785815738, −2.49187387919429110945340212541, −1.78717509160425731969564298014, −1.19986383983822357853496417162, 0,
1.19986383983822357853496417162, 1.78717509160425731969564298014, 2.49187387919429110945340212541, 3.01619245724457508796785815738, 3.56825199163943836512869689869, 4.22195688919300391079583105271, 4.83597441096562558631379203845, 5.42726382874489677927526159971, 6.03267634618817585971741043189, 6.45721627168504731854771604291, 6.88079171131385486535826064688, 7.39431735259013957516833938172, 7.70448810423057161566656504945, 8.230247022199406506458314543070
