| L(s) = 1 | − 6.72e4·49-s + 4.61e4·81-s − 1.28e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
| L(s) = 1 | − 4·49-s + 0.780·81-s − 8·121-s + 5.50e−6·127-s + 5.09e−6·131-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 + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s + 1.28e−6·227-s + 1.26e−6·229-s + 1.20e−6·233-s + 1.13e−6·239-s + 1.10e−6·241-s + 1.00e−6·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.248119324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248119324\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p^{5} T^{2} )^{4} \) |
| good | 3 | \( ( 1 - 23050 T^{4} + p^{20} T^{8} )^{2} \) |
| 5 | \( ( 1 + 14847382 T^{4} + p^{20} T^{8} )^{2} \) |
| 11 | \( ( 1 + p^{5} T^{2} )^{8} \) |
| 13 | \( ( 1 - 127006037450 T^{4} + p^{20} T^{8} )^{2} \) |
| 17 | \( ( 1 - p^{5} T^{2} )^{8} \) |
| 19 | \( ( 1 + 7721380741750 T^{4} + p^{20} T^{8} )^{2} \) |
| 23 | \( ( 1 - 11447050 T^{2} + p^{10} T^{4} )^{4} \) |
| 29 | \( ( 1 - p^{5} T^{2} )^{8} \) |
| 31 | \( ( 1 + p^{5} T^{2} )^{8} \) |
| 37 | \( ( 1 - p^{5} T^{2} )^{8} \) |
| 41 | \( ( 1 - p^{5} T^{2} )^{8} \) |
| 43 | \( ( 1 + p^{5} T^{2} )^{8} \) |
| 47 | \( ( 1 + p^{5} T^{2} )^{8} \) |
| 53 | \( ( 1 - p^{5} T^{2} )^{8} \) |
| 59 | \( ( 1 - 744013888469628650 T^{4} + p^{20} T^{8} )^{2} \) |
| 61 | \( ( 1 - 1094786861370139850 T^{4} + p^{20} T^{8} )^{2} \) |
| 67 | \( ( 1 + p^{5} T^{2} )^{8} \) |
| 71 | \( ( 1 - 3466330450 T^{2} + p^{10} T^{4} )^{4} \) |
| 73 | \( ( 1 - p^{5} T^{2} )^{8} \) |
| 79 | \( ( 1 + 6110532350 T^{2} + p^{10} T^{4} )^{4} \) |
| 83 | \( ( 1 - 969193474645656650 T^{4} + p^{20} T^{8} )^{2} \) |
| 89 | \( ( 1 - p^{5} T^{2} )^{8} \) |
| 97 | \( ( 1 - p^{5} T^{2} )^{8} \) |
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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
−4.00103368258608822698437648364, −3.82467229499468812202690802170, −3.76164584726329649867410439731, −3.67164028610364614404136124257, −3.39963122389543602156709593510, −3.30253465555663276078626219835, −3.25545646741368795123605564583, −2.88956747698985400445505299789, −2.66384646883662154769308807057, −2.59852233442923066312362535089, −2.53375301820023582801466529112, −2.48504249827391175073663955164, −2.26117313472129177686432858615, −2.20589740431948556926925940608, −1.58308800077942139889851197799, −1.51961109395418917152483123828, −1.48855182350296490650956284893, −1.40478185117529587840756293936, −1.32324697065554202171043164138, −1.22250982707846041753108743106, −0.800671000374596759658150203099, −0.53243724119594172235383423152, −0.36154024611582271843506437362, −0.27245827878928202700080934511, −0.096379374944114593314132179019,
0.096379374944114593314132179019, 0.27245827878928202700080934511, 0.36154024611582271843506437362, 0.53243724119594172235383423152, 0.800671000374596759658150203099, 1.22250982707846041753108743106, 1.32324697065554202171043164138, 1.40478185117529587840756293936, 1.48855182350296490650956284893, 1.51961109395418917152483123828, 1.58308800077942139889851197799, 2.20589740431948556926925940608, 2.26117313472129177686432858615, 2.48504249827391175073663955164, 2.53375301820023582801466529112, 2.59852233442923066312362535089, 2.66384646883662154769308807057, 2.88956747698985400445505299789, 3.25545646741368795123605564583, 3.30253465555663276078626219835, 3.39963122389543602156709593510, 3.67164028610364614404136124257, 3.76164584726329649867410439731, 3.82467229499468812202690802170, 4.00103368258608822698437648364
Plot not available for L-functions of degree greater than 10.
