| L(s) = 1 | − 2·3-s + 3·9-s − 4·17-s + 8·19-s + 2·25-s − 4·27-s − 12·41-s − 8·43-s + 2·49-s + 8·51-s − 16·57-s + 8·67-s − 4·73-s − 4·75-s + 5·81-s − 8·83-s − 4·89-s − 28·97-s − 4·113-s + 121-s + 24·123-s + 127-s + 16·129-s + 131-s + 137-s + 139-s − 4·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 0.970·17-s + 1.83·19-s + 2/5·25-s − 0.769·27-s − 1.87·41-s − 1.21·43-s + 2/7·49-s + 1.12·51-s − 2.11·57-s + 0.977·67-s − 0.468·73-s − 0.461·75-s + 5/9·81-s − 0.878·83-s − 0.423·89-s − 2.84·97-s − 0.376·113-s + 1/11·121-s + 2.16·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.329·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 557568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 557568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.290117207139140663833318098997, −7.69391773923694882195073682408, −7.09295558012764762645016653330, −6.84770563927529729400099349145, −6.53321046829649208757438762862, −5.77551802702969776640843012651, −5.46772010123763766734331991050, −5.01343088826377396495607521367, −4.60600194176777953268823364286, −3.95377687773827376614397292449, −3.35633617981568188254546704200, −2.74771027393953065821924272718, −1.80116474266194128879923496565, −1.13221342532811273007977656005, 0,
1.13221342532811273007977656005, 1.80116474266194128879923496565, 2.74771027393953065821924272718, 3.35633617981568188254546704200, 3.95377687773827376614397292449, 4.60600194176777953268823364286, 5.01343088826377396495607521367, 5.46772010123763766734331991050, 5.77551802702969776640843012651, 6.53321046829649208757438762862, 6.84770563927529729400099349145, 7.09295558012764762645016653330, 7.69391773923694882195073682408, 8.290117207139140663833318098997
