| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 9-s + 4·10-s + 4·13-s − 4·16-s − 2·18-s − 4·19-s − 4·20-s − 4·23-s − 3·25-s − 8·26-s + 4·29-s + 8·32-s + 2·36-s + 8·38-s + 8·41-s − 2·45-s + 8·46-s − 8·47-s − 2·49-s + 6·50-s + 8·52-s − 8·58-s − 8·64-s − 8·65-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 1/3·9-s + 1.26·10-s + 1.10·13-s − 16-s − 0.471·18-s − 0.917·19-s − 0.894·20-s − 0.834·23-s − 3/5·25-s − 1.56·26-s + 0.742·29-s + 1.41·32-s + 1/3·36-s + 1.29·38-s + 1.24·41-s − 0.298·45-s + 1.17·46-s − 1.16·47-s − 2/7·49-s + 0.848·50-s + 1.10·52-s − 1.05·58-s − 64-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
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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
−8.835479864989465913987076868701, −8.418026169715906351806134843849, −8.053385201781536183370105357260, −7.80427654007147735570937887834, −7.16801499136492885810138173228, −6.66439934046170059466176084573, −6.21527200903403420076786959518, −5.63272320625226366681911909037, −4.70641583168899826307168508447, −4.17367378557307136660967269093, −3.83435136506748562455497400428, −2.89585409207672243845095516787, −2.00467198200081980071181091200, −1.21167943138867662287721810874, 0,
1.21167943138867662287721810874, 2.00467198200081980071181091200, 2.89585409207672243845095516787, 3.83435136506748562455497400428, 4.17367378557307136660967269093, 4.70641583168899826307168508447, 5.63272320625226366681911909037, 6.21527200903403420076786959518, 6.66439934046170059466176084573, 7.16801499136492885810138173228, 7.80427654007147735570937887834, 8.053385201781536183370105357260, 8.418026169715906351806134843849, 8.835479864989465913987076868701
