| L(s) = 1 | − 3·5-s − 3·7-s + 6·11-s − 3·13-s − 3·19-s + 6·23-s − 15·29-s + 3·31-s + 9·35-s − 3·37-s − 6·41-s − 3·43-s + 15·47-s + 6·49-s − 18·53-s − 18·55-s + 3·59-s − 6·61-s + 9·65-s + 6·67-s + 15·71-s + 9·73-s − 18·77-s − 3·79-s + 18·83-s + 6·89-s + 9·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s + 1.80·11-s − 0.832·13-s − 0.688·19-s + 1.25·23-s − 2.78·29-s + 0.538·31-s + 1.52·35-s − 0.493·37-s − 0.937·41-s − 0.457·43-s + 2.18·47-s + 6/7·49-s − 2.47·53-s − 2.42·55-s + 0.390·59-s − 0.768·61-s + 1.11·65-s + 0.733·67-s + 1.78·71-s + 1.05·73-s − 2.05·77-s − 0.337·79-s + 1.97·83-s + 0.635·89-s + 0.943·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.827076230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827076230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 + 3 T + 9 T^{2} + 21 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.d_j_v |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 123 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ag_bk_aet |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) | 3.13.d_bq_db |
| 17 | $S_4\times C_2$ | \( 1 + 18 T^{2} + 9 T^{3} + 18 p T^{4} + p^{3} T^{6} \) | 3.17.a_s_j |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 65 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.d_v_cn |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 54 T^{2} - 177 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_cc_agv |
| 29 | $S_4\times C_2$ | \( 1 + 15 T + 135 T^{2} + 807 T^{3} + 135 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.p_ff_bfb |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 133 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_p_afd |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.d_bh_el |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 126 T^{2} + 483 T^{3} + 126 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.g_ew_sp |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 21 T^{2} - 205 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.d_v_ahx |
| 47 | $S_4\times C_2$ | \( 1 - 15 T + 177 T^{2} - 1329 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ap_gv_abzd |
| 53 | $S_4\times C_2$ | \( 1 + 18 T + 198 T^{2} + 1521 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.s_hq_cgn |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 435 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ad_et_aqt |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 138 T^{2} + 763 T^{3} + 138 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.g_fi_bdj |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 174 T^{2} - 745 T^{3} + 174 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_gs_abcr |
| 71 | $S_4\times C_2$ | \( 1 - 15 T + 231 T^{2} - 1833 T^{3} + 231 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ap_ix_acsn |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1235 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_hz_abvn |
| 79 | $S_4\times C_2$ | \( 1 + 3 T + 3 T^{2} - 889 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_d_abif |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 324 T^{2} - 3015 T^{3} + 324 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.as_mm_aelz |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 72 T^{2} + 21 T^{3} + 72 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ag_cu_v |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 309 T^{2} + 2887 T^{3} + 309 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.p_lx_ehb |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.85506079772465062265222206887, −6.60350446367447337779851587847, −6.45721555531639437052036236300, −6.33761861992199912818364879029, −5.92276239123946319178184153994, −5.69499919930806367504630143427, −5.42840296532811000776675277805, −5.09702186172703594020353339098, −5.05613830224751221703389139857, −4.71748986490699806248551283726, −4.20957137784876080958871830688, −4.18262572371607993125353717731, −3.98771467792027387238985254621, −3.64798517859511468129960396510, −3.53246356371530170911733496062, −3.41875260935524702034481904899, −2.95185028343729748333326031711, −2.64121542036339276213768635605, −2.50449904424520671888629234666, −1.90206296196706045423485109246, −1.64868520483827333505506150456, −1.63119978449273265167387007500, −0.886733606484407846689839335532, −0.44730010594066034148702603379, −0.38949969035053000853888353676,
0.38949969035053000853888353676, 0.44730010594066034148702603379, 0.886733606484407846689839335532, 1.63119978449273265167387007500, 1.64868520483827333505506150456, 1.90206296196706045423485109246, 2.50449904424520671888629234666, 2.64121542036339276213768635605, 2.95185028343729748333326031711, 3.41875260935524702034481904899, 3.53246356371530170911733496062, 3.64798517859511468129960396510, 3.98771467792027387238985254621, 4.18262572371607993125353717731, 4.20957137784876080958871830688, 4.71748986490699806248551283726, 5.05613830224751221703389139857, 5.09702186172703594020353339098, 5.42840296532811000776675277805, 5.69499919930806367504630143427, 5.92276239123946319178184153994, 6.33761861992199912818364879029, 6.45721555531639437052036236300, 6.60350446367447337779851587847, 6.85506079772465062265222206887
