L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s − 3·9-s + 2·12-s + 16-s − 3·18-s − 2·19-s + 2·24-s − 10·25-s − 14·27-s + 32-s − 3·36-s − 2·38-s − 2·43-s + 2·48-s − 13·49-s − 10·50-s − 14·54-s − 4·57-s + 12·59-s + 64-s + 16·67-s − 3·72-s − 2·73-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s − 9-s + 0.577·12-s + 1/4·16-s − 0.707·18-s − 0.458·19-s + 0.408·24-s − 2·25-s − 2.69·27-s + 0.176·32-s − 1/2·36-s − 0.324·38-s − 0.304·43-s + 0.288·48-s − 1.85·49-s − 1.41·50-s − 1.90·54-s − 0.529·57-s + 1.56·59-s + 1/8·64-s + 1.95·67-s − 0.353·72-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 645248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−8.155283673687832855649623537912, −7.79510581397524982652968495375, −7.43796356539382262615015727330, −6.52013654011009885219318723813, −6.45900320635304245713021356815, −5.79481497785218672150846122983, −5.30767778357765891574349430596, −5.02377036581249468373225138878, −4.04396073525979748019485118827, −3.77653001984849480071749037204, −3.36885759731685286955180224788, −2.51874373869269853485475383946, −2.41057075071613019915362974603, −1.59657674380372298540849156816, 0,
1.59657674380372298540849156816, 2.41057075071613019915362974603, 2.51874373869269853485475383946, 3.36885759731685286955180224788, 3.77653001984849480071749037204, 4.04396073525979748019485118827, 5.02377036581249468373225138878, 5.30767778357765891574349430596, 5.79481497785218672150846122983, 6.45900320635304245713021356815, 6.52013654011009885219318723813, 7.43796356539382262615015727330, 7.79510581397524982652968495375, 8.155283673687832855649623537912