| L(s) = 1 | + 2·3-s − 4·7-s + 2·9-s − 6·11-s + 4·13-s + 6·19-s − 8·21-s + 12·23-s − 4·27-s − 8·29-s − 4·31-s − 12·33-s − 16·37-s + 8·39-s + 4·43-s + 4·49-s + 12·57-s + 8·59-s + 12·61-s − 8·63-s − 30·67-s + 24·69-s + 40·73-s + 24·77-s + 4·79-s − 3·81-s − 22·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.51·7-s + 2/3·9-s − 1.80·11-s + 1.10·13-s + 1.37·19-s − 1.74·21-s + 2.50·23-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 + 4/7·49-s + 1.58·57-s + 1.04·59-s + 1.53·61-s − 1.00·63-s − 3.66·67-s + 2.88·69-s + 4.68·73-s + 2.73·77-s + 0.450·79-s − 1/3·81-s − 2.41·83-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
| \(L(1)\) |
\(\approx\) |
\(0.1482861751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1482861751\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( ( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 763 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 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 | $D_{4}$ | \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $D_{4}$ | \( ( 1 - 20 T + 235 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.76217304256824510487475551852, −6.74020335000088255489408589652, −6.27393995775745728453853557229, −6.08475244694134537901759854301, −5.77308144390051803854184154127, −5.57290568211868729647569139545, −5.26091487679869548973800404424, −5.25543215567807033855129600126, −5.12563242092986536868624905129, −4.92562980732840772962758132746, −4.42403622520968606139709588856, −4.00107790770485274444010507912, −3.89665929486561085081879086655, −3.70923452123494878696809798781, −3.43459648962968970039992280655, −3.33630793345540632330693706997, −2.84475279641438534551156746270, −2.83765063475230674589161042241, −2.82377078711959572211134999399, −2.07321895732660729742195889677, −2.04172371008821753623442869207, −1.70534784476808673439141319584, −1.02070425431454054163475831859, −1.00862993681028043417725593244, −0.06740925567190375817104291589,
0.06740925567190375817104291589, 1.00862993681028043417725593244, 1.02070425431454054163475831859, 1.70534784476808673439141319584, 2.04172371008821753623442869207, 2.07321895732660729742195889677, 2.82377078711959572211134999399, 2.83765063475230674589161042241, 2.84475279641438534551156746270, 3.33630793345540632330693706997, 3.43459648962968970039992280655, 3.70923452123494878696809798781, 3.89665929486561085081879086655, 4.00107790770485274444010507912, 4.42403622520968606139709588856, 4.92562980732840772962758132746, 5.12563242092986536868624905129, 5.25543215567807033855129600126, 5.26091487679869548973800404424, 5.57290568211868729647569139545, 5.77308144390051803854184154127, 6.08475244694134537901759854301, 6.27393995775745728453853557229, 6.74020335000088255489408589652, 6.76217304256824510487475551852
