| L(s) = 1 | − 4·4-s − 7-s − 6·11-s + 12·16-s − 25-s + 4·28-s − 12·29-s + 4·37-s − 2·43-s + 24·44-s − 6·49-s − 24·53-s − 32·64-s − 8·67-s − 12·71-s + 6·77-s + 16·79-s + 4·100-s + 36·107-s − 32·109-s − 12·112-s − 12·113-s + 48·116-s + 5·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.377·7-s − 1.80·11-s + 3·16-s − 1/5·25-s + 0.755·28-s − 2.22·29-s + 0.657·37-s − 0.304·43-s + 3.61·44-s − 6/7·49-s − 3.29·53-s − 4·64-s − 0.977·67-s − 1.42·71-s + 0.683·77-s + 1.80·79-s + 2/5·100-s + 3.48·107-s − 3.06·109-s − 1.13·112-s − 1.12·113-s + 4.45·116-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−7.62975310031092277352016304124, −7.43141403312000422638829977686, −6.42297241444199248506877762555, −6.13310834895975412778682267446, −5.51097905283736056328679977781, −5.27242056557243294239596962022, −4.85247022133919597468066156563, −4.41466206651142691425988858089, −3.91428937300881847314913370729, −3.23857552173567549926694025031, −3.09916574127981059996506876981, −2.12205923812240542124641474979, −1.30843197691730634587128188023, 0, 0,
1.30843197691730634587128188023, 2.12205923812240542124641474979, 3.09916574127981059996506876981, 3.23857552173567549926694025031, 3.91428937300881847314913370729, 4.41466206651142691425988858089, 4.85247022133919597468066156563, 5.27242056557243294239596962022, 5.51097905283736056328679977781, 6.13310834895975412778682267446, 6.42297241444199248506877762555, 7.43141403312000422638829977686, 7.62975310031092277352016304124
