| L(s) = 1 | − 240·17-s + 328·25-s + 816·41-s − 536·49-s − 1.93e3·73-s − 7.34e3·89-s − 2.96e3·97-s + 7.92e3·113-s + 8.02e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 296·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.42·17-s + 2.62·25-s + 3.10·41-s − 1.56·49-s − 3.10·73-s − 8.74·89-s − 3.09·97-s + 6.59·113-s + 6.02·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.134·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1330192525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1330192525\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 164 T^{2} + 19542 T^{4} - 164 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 7 | \( ( 1 + 268 T^{2} - 41658 T^{4} + 268 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4012 T^{2} + 7272246 T^{4} - 4012 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 13 | \( ( 1 - 148 T^{2} + 3687126 T^{4} - 148 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 + 60 T + 6118 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 19 | \( ( 1 - 17932 T^{2} + 171826710 T^{4} - 17932 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 - 4900 T^{2} + 206522790 T^{4} - 4900 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 29 | \( ( 1 - 56516 T^{2} + 1986668214 T^{4} - 56516 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 63916 T^{2} + 2595558 p^{2} T^{4} + 63916 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 37 | \( ( 1 - 95476 T^{2} + 7028163510 T^{4} - 95476 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 41 | \( ( 1 - 204 T + 806 p T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 - 273964 T^{2} + 31085635254 T^{4} - 273964 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 47 | \( ( 1 + 112892 T^{2} + 23215757766 T^{4} + 112892 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 53 | \( ( 1 - 507620 T^{2} + 107623720470 T^{4} - 507620 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 59 | \( ( 1 - 121324 T^{2} - 26790709386 T^{4} - 121324 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 + 132332 T^{2} + 107394800406 T^{4} + 132332 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 - 364108 T^{2} + 112096368726 T^{4} - 364108 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 71 | \( ( 1 + 4612 p T^{2} + 275267100390 T^{4} + 4612 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 484 T + 541686 T^{2} + 484 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 79 | \( ( 1 + 932332 T^{2} + 435081520806 T^{4} + 932332 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 83 | \( ( 1 - 539404 T^{2} + 296977663254 T^{4} - 539404 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 + 1836 T + 2086774 T^{2} + 1836 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 + 740 T + 1943814 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.73383295787935081753443259970, −3.71310039081462585177292609242, −3.46750259161953231436724651718, −3.35637100326392822492134091137, −3.09723682831516419102028626658, −2.92298887870634147017861010240, −2.85243219494961954132908540998, −2.85077827660327784830161955048, −2.74695332104695814375032805201, −2.69994288669671897950507309541, −2.56640593021480509127586301485, −2.13809632665568597118602319348, −2.01149314010461355207566147560, −1.95623543902953692579732440371, −1.93727046918475540583331458278, −1.69051032914352187361069855606, −1.49508914497466477179276019444, −1.31750218326804521209734162842, −1.16329946430623431297983349548, −1.03831991580905578803413510145, −0.74132435380500799694113562094, −0.59200752991177752465189296636, −0.55318733233634350709028126579, −0.18704147047753559682033196515, −0.03387559755214427791727877428,
0.03387559755214427791727877428, 0.18704147047753559682033196515, 0.55318733233634350709028126579, 0.59200752991177752465189296636, 0.74132435380500799694113562094, 1.03831991580905578803413510145, 1.16329946430623431297983349548, 1.31750218326804521209734162842, 1.49508914497466477179276019444, 1.69051032914352187361069855606, 1.93727046918475540583331458278, 1.95623543902953692579732440371, 2.01149314010461355207566147560, 2.13809632665568597118602319348, 2.56640593021480509127586301485, 2.69994288669671897950507309541, 2.74695332104695814375032805201, 2.85077827660327784830161955048, 2.85243219494961954132908540998, 2.92298887870634147017861010240, 3.09723682831516419102028626658, 3.35637100326392822492134091137, 3.46750259161953231436724651718, 3.71310039081462585177292609242, 3.73383295787935081753443259970
Plot not available for L-functions of degree greater than 10.
