| L(s) = 1 | + 2·2-s − 4·3-s + 2·4-s + 4·5-s − 8·6-s + 4·8-s + 11·9-s + 8·10-s − 6·11-s − 8·12-s + 12·13-s − 16·15-s + 8·16-s + 6·17-s + 22·18-s − 12·19-s + 8·20-s − 12·22-s + 12·23-s − 16·24-s + 5·25-s + 24·26-s − 20·27-s + 4·29-s − 32·30-s − 18·31-s + 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 2.30·3-s + 4-s + 1.78·5-s − 3.26·6-s + 1.41·8-s + 11/3·9-s + 2.52·10-s − 1.80·11-s − 2.30·12-s + 3.32·13-s − 4.13·15-s + 2·16-s + 1.45·17-s + 5.18·18-s − 2.75·19-s + 1.78·20-s − 2.55·22-s + 2.50·23-s − 3.26·24-s + 25-s + 4.70·26-s − 3.84·27-s + 0.742·29-s − 5.84·30-s − 3.23·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.587086522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.587086522\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 4 T + 5 T^{2} - 4 T^{3} - 20 T^{4} - 4 p T^{5} + 5 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 144 T^{3} + 587 T^{4} + 144 p T^{5} + 36 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 45 T^{2} + 84 T^{3} - 1276 T^{4} + 84 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 47 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 18 T + 172 T^{2} + 1152 T^{3} + 6483 T^{4} + 1152 p T^{5} + 172 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 36 T^{2} - 300 T^{3} + 551 T^{4} - 300 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 276 T^{3} + 4223 T^{4} + 276 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 90 T^{2} + 528 T^{3} - 8377 T^{4} + 528 p T^{5} + 90 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 24 T + 180 T^{2} + 204 T^{3} - 9145 T^{4} + 204 p T^{5} + 180 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 24 T + 6 p T^{2} + 3888 T^{3} + 34091 T^{4} + 3888 p T^{5} + 6 p^{3} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 47 T^{2} - 234 T^{3} + 972 T^{4} - 234 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 117 T^{2} - 996 T^{3} + 7052 T^{4} - 996 p T^{5} + 117 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 24 T + 274 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 26 T + 290 T^{2} + 1776 T^{3} + 10223 T^{4} + 1776 p T^{5} + 290 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T + 362 T^{2} - 4080 T^{3} + 37827 T^{4} - 4080 p T^{5} + 362 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 84 T^{3} - 4369 T^{4} + 84 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 281 T^{2} - 2808 T^{3} + 27840 T^{4} - 2808 p T^{5} + 281 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 7008 T^{3} + 81038 T^{4} + 7008 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.615280183517299004547206444701, −8.557323382509731521385788377108, −7.84935031506037839268252251067, −7.69467309878606273163091278089, −7.63441064527001352550785499993, −7.13965209944383900407490139829, −6.75964558561690298689410365857, −6.50315019048209312814684215982, −6.43510943479002154401116152804, −6.13738463311514541944696704096, −5.94309050916340196584876256897, −5.54081296054710821258962832805, −5.40449350856666590852768099944, −5.25066392289076861408707400053, −4.97167420324526224358655251553, −4.83948654018082342882956457567, −4.27377721080328117090664575825, −3.87893084131804322810833121941, −3.61779611779317045772102707530, −3.57274811013809362382744232887, −2.76306059388160278196055136554, −2.18825844411058816218370942461, −1.91572604091738877993461344532, −1.22681745046003708110200571109, −1.06518953540043698490977679152,
1.06518953540043698490977679152, 1.22681745046003708110200571109, 1.91572604091738877993461344532, 2.18825844411058816218370942461, 2.76306059388160278196055136554, 3.57274811013809362382744232887, 3.61779611779317045772102707530, 3.87893084131804322810833121941, 4.27377721080328117090664575825, 4.83948654018082342882956457567, 4.97167420324526224358655251553, 5.25066392289076861408707400053, 5.40449350856666590852768099944, 5.54081296054710821258962832805, 5.94309050916340196584876256897, 6.13738463311514541944696704096, 6.43510943479002154401116152804, 6.50315019048209312814684215982, 6.75964558561690298689410365857, 7.13965209944383900407490139829, 7.63441064527001352550785499993, 7.69467309878606273163091278089, 7.84935031506037839268252251067, 8.557323382509731521385788377108, 8.615280183517299004547206444701
