L(s) = 1 | + 12·5-s + 5·9-s + 70·25-s + 60·45-s + 28·49-s − 24·53-s + 16·81-s + 68·97-s + 22·121-s + 240·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | + 5.36·5-s + 5/3·9-s + 14·25-s + 8.94·45-s + 4·49-s − 3.29·53-s + 16/9·81-s + 6.90·97-s + 2·121-s + 21.4·125-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 + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(25.18529953\) |
\(L(\frac12)\) |
\(\approx\) |
\(25.18529953\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 107 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2}( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.27262877725639534841066011366, −6.22250452430175239972304068500, −6.01052717103048903051056508740, −5.93436466935571743994124194222, −5.85121674991451863576613428567, −5.34368273453212382061853898875, −5.31027211201273231573137939770, −5.05349295046920248260008110624, −4.97894433527103838839265933297, −4.71897021818221406725354986223, −4.44033260490343137021421261708, −4.05224804733156786097872002678, −4.04501852601550726656471845539, −3.42960181470317596326110441534, −3.37739581559481485973456106699, −2.97696186418289381191236037308, −2.73797337309425277775698518710, −2.26074205494870076190728034418, −2.13567559155893434849590118098, −2.10237995294368779675716806930, −1.98989279927646202130256757757, −1.54174575365456392411865595714, −1.25349564153593755171813508040, −1.13288604239763917716737707956, −0.64775156322734916349604737346,
0.64775156322734916349604737346, 1.13288604239763917716737707956, 1.25349564153593755171813508040, 1.54174575365456392411865595714, 1.98989279927646202130256757757, 2.10237995294368779675716806930, 2.13567559155893434849590118098, 2.26074205494870076190728034418, 2.73797337309425277775698518710, 2.97696186418289381191236037308, 3.37739581559481485973456106699, 3.42960181470317596326110441534, 4.04501852601550726656471845539, 4.05224804733156786097872002678, 4.44033260490343137021421261708, 4.71897021818221406725354986223, 4.97894433527103838839265933297, 5.05349295046920248260008110624, 5.31027211201273231573137939770, 5.34368273453212382061853898875, 5.85121674991451863576613428567, 5.93436466935571743994124194222, 6.01052717103048903051056508740, 6.22250452430175239972304068500, 6.27262877725639534841066011366