L(s) = 1 | − 2·2-s + 2·4-s − 3·9-s − 2·11-s − 4·16-s + 6·18-s + 4·22-s + 25-s + 8·32-s − 6·36-s + 12·43-s − 4·44-s − 7·49-s − 2·50-s − 8·64-s − 28·67-s − 24·86-s + 14·98-s + 6·99-s + 2·100-s − 4·107-s − 16·113-s − 19·121-s + 127-s + 131-s + 56·134-s + 137-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 9-s − 0.603·11-s − 16-s + 1.41·18-s + 0.852·22-s + 1/5·25-s + 1.41·32-s − 36-s + 1.82·43-s − 0.603·44-s − 49-s − 0.282·50-s − 64-s − 3.42·67-s − 2.58·86-s + 1.41·98-s + 0.603·99-s + 1/5·100-s − 0.386·107-s − 1.50·113-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 4.83·134-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
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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
−9.413833215018796083736788450286, −8.905620132103282237099470021544, −8.690318352415095273348099800880, −8.032256305048544541115493926269, −7.63572025887558713136221609863, −7.27326475782480216593475577334, −6.46570458456308613518881961491, −6.02378822140684312049628434559, −5.35555342291496416407146876793, −4.72232396451229340241889116709, −3.99977094286265568450903155609, −2.96099201703757334117530089963, −2.47007791992779484158478737131, −1.37226864740468197156839338898, 0,
1.37226864740468197156839338898, 2.47007791992779484158478737131, 2.96099201703757334117530089963, 3.99977094286265568450903155609, 4.72232396451229340241889116709, 5.35555342291496416407146876793, 6.02378822140684312049628434559, 6.46570458456308613518881961491, 7.27326475782480216593475577334, 7.63572025887558713136221609863, 8.032256305048544541115493926269, 8.690318352415095273348099800880, 8.905620132103282237099470021544, 9.413833215018796083736788450286