| L(s) = 1 | − 4·3-s + 4·4-s − 4·7-s + 8·9-s − 4·11-s − 16·12-s − 4·13-s + 12·16-s + 16·21-s + 12·23-s + 8·25-s − 16·27-s − 16·28-s − 12·29-s + 16·33-s + 32·36-s − 16·37-s + 16·39-s + 24·41-s + 4·43-s − 16·44-s − 12·47-s − 48·48-s + 8·49-s − 16·52-s + 24·53-s − 32·63-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·4-s − 1.51·7-s + 8/3·9-s − 1.20·11-s − 4.61·12-s − 1.10·13-s + 3·16-s + 3.49·21-s + 2.50·23-s + 8/5·25-s − 3.07·27-s − 3.02·28-s − 2.22·29-s + 2.78·33-s + 16/3·36-s − 2.63·37-s + 2.56·39-s + 3.74·41-s + 0.609·43-s − 2.41·44-s − 1.75·47-s − 6.92·48-s + 8/7·49-s − 2.21·52-s + 3.29·53-s − 4.03·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8180683284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8180683284\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) | |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 16 T^{3} + 31 T^{4} + 16 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.e_i_q_bf |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - p^{2} T^{4} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.e_i_a_abx |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.13.e_i_ci_re |
| 17 | $C_2^3$ | \( 1 + 511 T^{4} + p^{4} T^{8} \) | 4.17.a_a_a_tr |
| 19 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 195 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_aw_a_hn |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 300 T^{3} + 1246 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.am_cu_alo_bvy |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) | 4.29.m_go_bsi_mfr |
| 31 | $D_4\times C_2$ | \( 1 - 70 T^{2} + 2499 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_acs_a_dsd |
| 37 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 960 T^{3} + 6671 T^{4} + 960 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.q_ey_bky_jwp |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ay_ng_afae_blgc |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 36 T^{3} - 994 T^{4} - 36 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ae_i_abk_abmg |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 12 T^{3} - 2114 T^{4} + 12 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.m_cu_m_addi |
| 53 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2976 T^{3} + 25711 T^{4} - 2976 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ay_lc_aekm_bmax |
| 59 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_abo_a_kxe |
| 61 | $C_2^2$ | \( ( 1 - 41 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ade_a_nmx |
| 67 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.i_bg_xc_qog |
| 71 | $D_4\times C_2$ | \( 1 - 262 T^{2} + 27171 T^{4} - 262 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_akc_a_bofb |
| 73 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1104 T^{3} + 9506 T^{4} - 1104 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.aq_ey_abqm_obq |
| 79 | $D_{4}$ | \( ( 1 - 8 T + 156 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.aq_om_afoq_crla |
| 83 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 12 T^{3} - 6722 T^{4} - 12 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.am_cu_am_ajyo |
| 89 | $D_4\times C_2$ | \( 1 - 322 T^{2} + 41475 T^{4} - 322 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_amk_a_cjjf |
| 97 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} + 6048 T^{3} + 62978 T^{4} + 6048 p T^{5} + 512 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.bg_ts_iyq_dpeg |
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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
−7.931633396903095695846631364442, −7.46210124702213924291097301270, −7.20782946847398819199395098269, −7.19916200967847453465243623949, −7.19804448178811703297467682983, −6.82069373224075595778422125663, −6.60734009808162621283696236969, −6.18439511339365955333205600969, −6.17808664728437831102184600082, −5.75700362132430157114396885682, −5.57617998532376994321799642371, −5.33755066075408757823266743010, −5.23622244934544616799439376480, −4.99101979060264720896093534054, −4.62661726029470666689237642456, −4.11031609402207857802147143876, −3.66617735173355927779263043828, −3.45153459148642940049731023141, −3.25646938997289953732155103063, −2.55854588223787873021944848576, −2.45199185410129594669612086348, −2.38194281337168651809457570807, −1.53365465953528568559309835450, −1.00765795669145838374407395148, −0.42954050213350876526836354597,
0.42954050213350876526836354597, 1.00765795669145838374407395148, 1.53365465953528568559309835450, 2.38194281337168651809457570807, 2.45199185410129594669612086348, 2.55854588223787873021944848576, 3.25646938997289953732155103063, 3.45153459148642940049731023141, 3.66617735173355927779263043828, 4.11031609402207857802147143876, 4.62661726029470666689237642456, 4.99101979060264720896093534054, 5.23622244934544616799439376480, 5.33755066075408757823266743010, 5.57617998532376994321799642371, 5.75700362132430157114396885682, 6.17808664728437831102184600082, 6.18439511339365955333205600969, 6.60734009808162621283696236969, 6.82069373224075595778422125663, 7.19804448178811703297467682983, 7.19916200967847453465243623949, 7.20782946847398819199395098269, 7.46210124702213924291097301270, 7.931633396903095695846631364442
