L(s) = 1 | + 4·3-s − 2·9-s − 4·11-s − 8·19-s − 40·27-s − 16·33-s + 24·41-s − 32·57-s − 32·59-s − 12·67-s − 24·73-s − 55·81-s − 20·83-s − 8·89-s − 28·97-s + 8·99-s + 8·107-s + 16·113-s + 8·121-s + 96·123-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 2/3·9-s − 1.20·11-s − 1.83·19-s − 7.69·27-s − 2.78·33-s + 3.74·41-s − 4.23·57-s − 4.16·59-s − 1.46·67-s − 2.80·73-s − 6.11·81-s − 2.19·83-s − 0.847·89-s − 2.84·97-s + 0.804·99-s + 0.773·107-s + 1.50·113-s + 8/11·121-s + 8.65·123-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5038997757\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5038997757\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 5 | $C_2^3$ | \( 1 - 41 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 97 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^3$ | \( 1 + 983 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 2143 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 9457 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 132 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.183503863471162482817469421664, −7.949970144349250383813649320626, −7.71442207920456603963892291435, −7.60721889227404273118802203977, −7.37186392693435805117818798283, −7.14328859976291820371974682354, −6.53306096055565267878808075195, −6.29858617194998800922041737282, −5.98747868371912356386435455814, −5.81671496112109025206349593107, −5.68900522642042334314257742942, −5.56051991924676673029888604411, −4.99132920166081112136667617576, −4.44603574149262605827057381643, −4.44283662951956588683922956516, −4.21350629763050392852135877784, −3.57496313961042972605135272492, −3.48804977272946765309609836342, −2.93739711415506004510821247999, −2.67146770059946876187202292332, −2.65575089887556229422964292453, −2.63960168777552721901613107640, −1.83594166905495058642333256099, −1.64159201182266667141281234590, −0.19767752283746219961372471541,
0.19767752283746219961372471541, 1.64159201182266667141281234590, 1.83594166905495058642333256099, 2.63960168777552721901613107640, 2.65575089887556229422964292453, 2.67146770059946876187202292332, 2.93739711415506004510821247999, 3.48804977272946765309609836342, 3.57496313961042972605135272492, 4.21350629763050392852135877784, 4.44283662951956588683922956516, 4.44603574149262605827057381643, 4.99132920166081112136667617576, 5.56051991924676673029888604411, 5.68900522642042334314257742942, 5.81671496112109025206349593107, 5.98747868371912356386435455814, 6.29858617194998800922041737282, 6.53306096055565267878808075195, 7.14328859976291820371974682354, 7.37186392693435805117818798283, 7.60721889227404273118802203977, 7.71442207920456603963892291435, 7.949970144349250383813649320626, 8.183503863471162482817469421664