| L(s) = 1 | + 6.29e4·13-s − 1.29e6·25-s + 4.61e6·37-s + 3.30e7·49-s − 2.63e7·61-s − 7.09e7·73-s − 1.17e8·97-s + 2.58e8·109-s + 7.47e8·121-s + ⋯ |
| L(s) = 1 | + 2.20·13-s − 3.30·25-s + 2.46·37-s + 5.74·49-s − 1.90·61-s − 2.49·73-s − 1.33·97-s + 1.83·109-s + 3.48·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+4)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(16.50023518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.50023518\) |
| \(L(5)\) |
|
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 | \( ( 1 + 129116 p T^{2} + 59762749854 p T^{4} + 129116 p^{17} T^{6} + p^{32} T^{8} )^{2} \) |
| 7 | \( ( 1 - 337702 p^{2} T^{2} + 53595603483 p^{4} T^{4} - 337702 p^{18} T^{6} + p^{32} T^{8} )^{2} \) |
| 11 | \( ( 1 - 373583644 T^{2} + 94482052446383286 T^{4} - 373583644 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 13 | \( ( 1 - 15734 T + 1267775511 T^{2} - 15734 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
| 17 | \( ( 1 + 14835789004 T^{2} + \)\(13\!\cdots\!86\)\( T^{4} + 14835789004 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 19 | \( ( 1 - 34475138878 T^{2} + \)\(80\!\cdots\!03\)\( T^{4} - 34475138878 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 23 | \( ( 1 - 293106482524 T^{2} + \)\(33\!\cdots\!66\)\( T^{4} - 293106482524 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 29 | \( ( 1 + 1163599046404 T^{2} + \)\(76\!\cdots\!66\)\( T^{4} + 1163599046404 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 31 | \( ( 1 - 3392286768388 T^{2} + \)\(43\!\cdots\!98\)\( T^{4} - 3392286768388 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 37 | \( ( 1 - 1153642 T + 3116932236663 T^{2} - 1153642 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
| 41 | \( ( 1 + 30520933013764 T^{2} + \)\(36\!\cdots\!26\)\( T^{4} + 30520933013764 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 43 | \( ( 1 - 25298154980548 T^{2} + \)\(41\!\cdots\!98\)\( T^{4} - 25298154980548 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 47 | \( ( 1 - 18104199732124 T^{2} + \)\(71\!\cdots\!66\)\( T^{4} - 18104199732124 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 53 | \( ( 1 + 90977328416164 T^{2} + \)\(64\!\cdots\!86\)\( T^{4} + 90977328416164 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 59 | \( ( 1 + 37268719470116 T^{2} + \)\(93\!\cdots\!46\)\( T^{4} + 37268719470116 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 61 | \( ( 1 + 6590222 T + 279430932499503 T^{2} + 6590222 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
| 67 | \( ( 1 - 1475857357130494 T^{2} + \)\(86\!\cdots\!71\)\( T^{4} - 1475857357130494 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 71 | \( ( 1 + 590799500699036 T^{2} + \)\(84\!\cdots\!46\)\( T^{4} + 590799500699036 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 73 | \( ( 1 + 17742262 T + 1302763992757323 T^{2} + 17742262 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
| 79 | \( ( 1 - 5170419875472838 T^{2} + \)\(11\!\cdots\!03\)\( T^{4} - 5170419875472838 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 83 | \( ( 1 - 8506104650715364 T^{2} + \)\(28\!\cdots\!66\)\( T^{4} - 8506104650715364 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 89 | \( ( 1 + 7034223797307724 T^{2} + \)\(38\!\cdots\!66\)\( T^{4} + 7034223797307724 p^{16} T^{6} + p^{32} T^{8} )^{2} \) |
| 97 | \( ( 1 + 29491102 T + 14355589936614243 T^{2} + 29491102 p^{8} T^{3} + p^{16} T^{4} )^{4} \) |
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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
−3.58236398350197620892209858772, −3.57225820750179717384955580975, −3.29459778681054992038572298540, −3.22244595015949699026909621750, −2.95153014810911563543053235314, −2.82993712070702779706147370727, −2.76490147711897242591244295200, −2.59157765368021761181841433932, −2.41725645960911433545507229048, −2.24968309524385023788156619095, −2.24696078697861556893068805093, −2.03888511312947927341854915862, −1.88082578195199575560986538852, −1.64284153165584641396469411344, −1.59316929950220137628149379627, −1.42397549302227371208414169796, −1.20519569398642241845956416528, −1.16660956669299255921846777684, −1.08434524740761072341868036030, −0.77990994546078836725698173895, −0.69977980582628064510898773360, −0.46885313335324301058715780958, −0.43769060709357196859906789125, −0.33403879898037056026987051942, −0.14714062985361292327879224810,
0.14714062985361292327879224810, 0.33403879898037056026987051942, 0.43769060709357196859906789125, 0.46885313335324301058715780958, 0.69977980582628064510898773360, 0.77990994546078836725698173895, 1.08434524740761072341868036030, 1.16660956669299255921846777684, 1.20519569398642241845956416528, 1.42397549302227371208414169796, 1.59316929950220137628149379627, 1.64284153165584641396469411344, 1.88082578195199575560986538852, 2.03888511312947927341854915862, 2.24696078697861556893068805093, 2.24968309524385023788156619095, 2.41725645960911433545507229048, 2.59157765368021761181841433932, 2.76490147711897242591244295200, 2.82993712070702779706147370727, 2.95153014810911563543053235314, 3.22244595015949699026909621750, 3.29459778681054992038572298540, 3.57225820750179717384955580975, 3.58236398350197620892209858772
Plot not available for L-functions of degree greater than 10.
