| L(s) = 1 | − 8·2-s − 2·3-s + 30·4-s + 10·5-s + 16·6-s − 80·8-s + 23·9-s − 80·10-s + 14·11-s − 60·12-s + 100·13-s − 20·15-s + 192·16-s + 50·17-s − 184·18-s − 36·19-s + 300·20-s − 112·22-s − 244·23-s + 160·24-s + 25·25-s − 800·26-s − 22·27-s − 52·29-s + 160·30-s + 120·31-s − 480·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 0.384·3-s + 15/4·4-s + 0.894·5-s + 1.08·6-s − 3.53·8-s + 0.851·9-s − 2.52·10-s + 0.383·11-s − 1.44·12-s + 2.13·13-s − 0.344·15-s + 3·16-s + 0.713·17-s − 2.40·18-s − 0.434·19-s + 3.35·20-s − 1.08·22-s − 2.21·23-s + 1.36·24-s + 1/5·25-s − 6.03·26-s − 0.156·27-s − 0.332·29-s + 0.973·30-s + 0.695·31-s − 2.65·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01136220751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01136220751\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{3} T + 17 p T^{2} + 7 p^{4} T^{3} + 81 p^{2} T^{4} + 7 p^{7} T^{5} + 17 p^{7} T^{6} + p^{12} T^{7} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 T - 19 T^{2} - 62 T^{3} - 308 T^{4} - 62 p^{3} T^{5} - 19 p^{6} T^{6} + 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 14 T - 467 T^{2} + 27986 T^{3} - 1592868 T^{4} + 27986 p^{3} T^{5} - 467 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 50 T + 4987 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 50 T - 4079 T^{2} + 9550 p T^{3} + 23748 p^{2} T^{4} + 9550 p^{4} T^{5} - 4079 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 36 T - 8874 T^{2} - 127728 T^{3} + 47493755 T^{4} - 127728 p^{3} T^{5} - 8874 p^{6} T^{6} + 36 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 244 T + 29566 T^{2} + 1375184 T^{3} + 25790499 T^{4} + 1375184 p^{3} T^{5} + 29566 p^{6} T^{6} + 244 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 26 T + 47795 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 120 T + 16018 T^{2} + 7344000 T^{3} - 1313876157 T^{4} + 7344000 p^{3} T^{5} + 16018 p^{6} T^{6} - 120 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 564 T + 144466 T^{2} + 40790736 T^{3} + 11469133803 T^{4} + 40790736 p^{3} T^{5} + 144466 p^{6} T^{6} + 564 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 p T + 133986 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 260 T + 166666 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 350 T - 80923 T^{2} + 1478050 T^{3} + 17883384100 T^{4} + 1478050 p^{3} T^{5} - 80923 p^{6} T^{6} - 350 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 56 T - 262634 T^{2} + 1791104 T^{3} + 48002453499 T^{4} + 1791104 p^{3} T^{5} - 262634 p^{6} T^{6} - 56 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 616 T + 174077 T^{2} - 616 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 336 T - 345962 T^{2} + 1645056 T^{3} + 133405042827 T^{4} + 1645056 p^{3} T^{5} - 345962 p^{6} T^{6} + 336 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 152 T - 576006 T^{2} + 367232 T^{3} + 261525581579 T^{4} + 367232 p^{3} T^{5} - 576006 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 952 T + p^{3} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 676 T - 198630 T^{2} - 82761328 T^{3} + 100713568355 T^{4} - 82761328 p^{3} T^{5} - 198630 p^{6} T^{6} + 676 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1014 T - 91923 T^{2} + 135917574 T^{3} + 504638390996 T^{4} + 135917574 p^{3} T^{5} - 91923 p^{6} T^{6} + 1014 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 376 T + 458918 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 216 T - 1371074 T^{2} - 1683072 T^{3} + 1480086028275 T^{4} - 1683072 p^{3} T^{5} - 1371074 p^{6} T^{6} - 216 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 2742 T + 3608187 T^{2} - 2742 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.539438678049514695525480060779, −8.444413408884437876337603695731, −7.976191526045241844615773476708, −7.71786567822794458414042360446, −7.21950498742737981642010798348, −7.12327141954820329546198569252, −7.09683468310378613981254684804, −6.38344228501129898677326301083, −6.35141312024303987606372214028, −6.04855542913715541680742446541, −5.81630160966429841900570768816, −5.63894440276911418992529312936, −5.04267602319919802724494576130, −4.90176051419986422871195769563, −4.17424417507332596701987388713, −4.11933555023545107173000631738, −3.45192419917778259636137806291, −3.42723471665700800535540436970, −2.98913705592743077068522950212, −2.01975662607299468224477918144, −1.96005136476061601856550669498, −1.52192939250796835000111180097, −1.33654565461070132021952488522, −0.74199361904856778040632325181, −0.04324080896232716535802287510,
0.04324080896232716535802287510, 0.74199361904856778040632325181, 1.33654565461070132021952488522, 1.52192939250796835000111180097, 1.96005136476061601856550669498, 2.01975662607299468224477918144, 2.98913705592743077068522950212, 3.42723471665700800535540436970, 3.45192419917778259636137806291, 4.11933555023545107173000631738, 4.17424417507332596701987388713, 4.90176051419986422871195769563, 5.04267602319919802724494576130, 5.63894440276911418992529312936, 5.81630160966429841900570768816, 6.04855542913715541680742446541, 6.35141312024303987606372214028, 6.38344228501129898677326301083, 7.09683468310378613981254684804, 7.12327141954820329546198569252, 7.21950498742737981642010798348, 7.71786567822794458414042360446, 7.976191526045241844615773476708, 8.444413408884437876337603695731, 8.539438678049514695525480060779
