| L(s) = 1 | − 2·3-s + 2·9-s − 6·11-s − 4·13-s − 8·17-s − 6·19-s + 4·27-s + 8·29-s − 4·31-s + 12·33-s + 16·37-s + 8·39-s + 4·43-s + 32·47-s + 4·49-s + 16·51-s + 12·57-s − 8·59-s + 12·61-s − 30·67-s − 4·79-s − 3·81-s + 22·83-s − 16·87-s + 8·93-s − 12·99-s + 8·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2/3·9-s − 1.80·11-s − 1.10·13-s − 1.94·17-s − 1.37·19-s + 0.769·27-s + 1.48·29-s − 0.718·31-s + 2.08·33-s + 2.63·37-s + 1.28·39-s + 0.609·43-s + 4.66·47-s + 4/7·49-s + 2.24·51-s + 1.58·57-s − 1.04·59-s + 1.53·61-s − 3.66·67-s − 0.450·79-s − 1/3·81-s + 2.41·83-s − 1.71·87-s + 0.829·93-s − 1.20·99-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.815905499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815905499\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 58 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 60 T^{3} + 199 T^{4} + 60 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 28 T^{3} - 302 T^{4} - 28 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 108 T^{3} + 647 T^{4} + 108 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1338 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 752 T^{3} + 4318 T^{4} - 752 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 57 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 4 p T^{3} - 2 p^{2} T^{4} + 4 p^{2} T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $C_2^3$ | \( 1 + 1438 T^{4} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 360 T^{3} + 3854 T^{4} + 360 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 108 T^{3} - 4738 T^{4} + 108 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 30 T + 450 T^{2} + 5220 T^{3} + 49103 T^{4} + 5220 p T^{5} + 450 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 18214 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 70 T^{2} + 7483 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 148 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 22 T + 242 T^{2} - 3036 T^{3} + 35063 T^{4} - 3036 p T^{5} + 242 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 86 T^{2} + 3435 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 150 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.80207046743745894820754360355, −6.29348190100805176253806152165, −6.22537114419871080106546209066, −6.14340612147722856785704049096, −5.76236224648033828942621041757, −5.67712232131631853341167579835, −5.47878770274666310305277103694, −5.33628574601262472958724163697, −4.66775943924797378787877386985, −4.60303298702409387919673704310, −4.57205598899801062307428613661, −4.56524291205268697569200895763, −4.27187274539724646253298654853, −3.98707257861592397247559777689, −3.45220310801943847499178651773, −3.28622693289934310391475457305, −2.76021430837972293409226783409, −2.74789839054856825815862546200, −2.41925888802632049721144860717, −2.22502577993279540396363817023, −2.04783949570786613810893349096, −1.64993656369309558021626856825, −0.73080639112088721636148502379, −0.62888714213558903279079969577, −0.53390163103972747278135730645,
0.53390163103972747278135730645, 0.62888714213558903279079969577, 0.73080639112088721636148502379, 1.64993656369309558021626856825, 2.04783949570786613810893349096, 2.22502577993279540396363817023, 2.41925888802632049721144860717, 2.74789839054856825815862546200, 2.76021430837972293409226783409, 3.28622693289934310391475457305, 3.45220310801943847499178651773, 3.98707257861592397247559777689, 4.27187274539724646253298654853, 4.56524291205268697569200895763, 4.57205598899801062307428613661, 4.60303298702409387919673704310, 4.66775943924797378787877386985, 5.33628574601262472958724163697, 5.47878770274666310305277103694, 5.67712232131631853341167579835, 5.76236224648033828942621041757, 6.14340612147722856785704049096, 6.22537114419871080106546209066, 6.29348190100805176253806152165, 6.80207046743745894820754360355
