| L(s) = 1 | − 58·7-s + 658·13-s − 3.59e3·19-s + 1.52e3·25-s − 1.04e4·31-s + 1.75e4·37-s − 3.99e4·43-s − 3.10e4·49-s − 2.13e3·61-s + 1.24e5·67-s − 9.61e4·73-s − 9.99e4·79-s − 3.81e4·91-s + 2.58e4·97-s + 1.55e5·103-s − 3.54e4·109-s − 3.14e5·121-s + 127-s + 131-s + 2.08e5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.447·7-s + 1.07·13-s − 2.28·19-s + 0.488·25-s − 1.95·31-s + 2.10·37-s − 3.29·43-s − 1.84·49-s − 0.0735·61-s + 3.37·67-s − 2.11·73-s − 1.80·79-s − 0.483·91-s + 0.278·97-s + 1.43·103-s − 0.285·109-s − 1.95·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 1.02·133-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 1526 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 29 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 314326 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 329 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2020286 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1799 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 198770 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 39031642 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 5228 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8783 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9156974 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 19976 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 341046910 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31068470 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 1396994998 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 1069 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 62077 T + p^{5} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 1457026126 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 48079 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 49979 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4552161670 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3438704914 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12917 T + p^{5} T^{2} )^{2} \) |
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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
−9.959008791504749717115376178157, −9.921680760191217507101228494791, −9.033099714708472991791959031873, −8.823786333568887481568227232919, −8.309280815464786728621991510980, −7.991320351214423259599279268354, −7.29402252072290441585309283025, −6.70043190702282538817602065796, −6.24470362644595188833599859321, −6.19609318208325796254962320145, −5.21072922587983524318233197779, −4.90235737345485749076382477639, −4.01164845863122720206074756180, −3.82306207866799221579569622316, −3.11277602618811378087438493996, −2.43096881053732151550753689623, −1.74348988215905730435181456729, −1.21145057919366873065850876229, 0, 0,
1.21145057919366873065850876229, 1.74348988215905730435181456729, 2.43096881053732151550753689623, 3.11277602618811378087438493996, 3.82306207866799221579569622316, 4.01164845863122720206074756180, 4.90235737345485749076382477639, 5.21072922587983524318233197779, 6.19609318208325796254962320145, 6.24470362644595188833599859321, 6.70043190702282538817602065796, 7.29402252072290441585309283025, 7.991320351214423259599279268354, 8.309280815464786728621991510980, 8.823786333568887481568227232919, 9.033099714708472991791959031873, 9.921680760191217507101228494791, 9.959008791504749717115376178157
