| L(s) = 1 | + 7-s − 5·9-s + 6·11-s − 12·23-s + 25-s − 18·29-s − 20·37-s + 4·43-s + 49-s − 5·63-s + 16·67-s + 6·77-s + 10·79-s + 16·81-s − 30·99-s − 12·107-s − 38·109-s − 12·113-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 12·161-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 5/3·9-s + 1.80·11-s − 2.50·23-s + 1/5·25-s − 3.34·29-s − 3.28·37-s + 0.609·43-s + 1/7·49-s − 0.629·63-s + 1.95·67-s + 0.683·77-s + 1.12·79-s + 16/9·81-s − 3.01·99-s − 1.16·107-s − 3.63·109-s − 1.12·113-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.945·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−8.980861688416760643046869733716, −8.796877372751552773865430357179, −8.046356379591346049691472796650, −7.896967859524111771416111170055, −7.00666421055054313159273263629, −6.67050874393051432711601966688, −6.01984510959178463345123026067, −5.41652951484887900961612326217, −5.40018100386713622724414679311, −4.13702557269363707764178227582, −3.80030272906745320072379387841, −3.35587936623394959029150485777, −2.14587146010097196258543648767, −1.73203110056877278278191340704, 0,
1.73203110056877278278191340704, 2.14587146010097196258543648767, 3.35587936623394959029150485777, 3.80030272906745320072379387841, 4.13702557269363707764178227582, 5.40018100386713622724414679311, 5.41652951484887900961612326217, 6.01984510959178463345123026067, 6.67050874393051432711601966688, 7.00666421055054313159273263629, 7.896967859524111771416111170055, 8.046356379591346049691472796650, 8.796877372751552773865430357179, 8.980861688416760643046869733716
