| L(s) = 1 | + 4·3-s + 6·5-s + 2·7-s + 13·9-s + 11-s + 4·13-s + 24·15-s + 2·17-s + 5·19-s + 8·21-s − 4·23-s + 25·25-s + 30·27-s − 10·29-s − 2·31-s + 4·33-s + 12·35-s + 18·37-s + 16·39-s − 2·41-s − 6·43-s + 78·45-s − 8·47-s + 7·49-s + 8·51-s + 4·53-s + 6·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2.68·5-s + 0.755·7-s + 13/3·9-s + 0.301·11-s + 1.10·13-s + 6.19·15-s + 0.485·17-s + 1.14·19-s + 1.74·21-s − 0.834·23-s + 5·25-s + 5.77·27-s − 1.85·29-s − 0.359·31-s + 0.696·33-s + 2.02·35-s + 2.95·37-s + 2.56·39-s − 0.312·41-s − 0.914·43-s + 11.6·45-s − 1.16·47-s + 49-s + 1.12·51-s + 0.549·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.15964057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.15964057\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $C_2^2:C_4$ | \( 1 - 4 T + p T^{2} + 10 T^{3} - 29 T^{4} + 10 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ae_d_k_abd |
| 5 | $C_2^2:C_4$ | \( 1 - 6 T + 11 T^{2} - 6 T^{3} + T^{4} - 6 p T^{5} + 11 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ag_l_ag_b |
| 7 | $C_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 20 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ac_ad_u_at |
| 13 | $C_2^2:C_4$ | \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ae_d_aby_nd |
| 17 | $C_2^2:C_4$ | \( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 20 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ac_an_au_nd |
| 19 | $C_2^2:C_4$ | \( 1 - 5 T + 21 T^{2} - 145 T^{3} + 956 T^{4} - 145 p T^{5} + 21 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.af_v_afp_bku |
| 23 | $D_{4}$ | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.e_dk_ka_elq |
| 29 | $C_4\times C_2$ | \( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 200 p T^{5} + 31 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.k_bf_hs_csb |
| 31 | $C_4\times C_2$ | \( 1 + 2 T - 27 T^{2} - 116 T^{3} + 605 T^{4} - 116 p T^{5} - 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.c_abb_aem_xh |
| 37 | $C_2^2:C_4$ | \( 1 - 18 T + 107 T^{2} - 210 T^{3} + T^{4} - 210 p T^{5} + 107 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.as_ed_aic_b |
| 41 | $C_2^2:C_4$ | \( 1 + 2 T - 17 T^{2} - 236 T^{3} + 525 T^{4} - 236 p T^{5} - 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.c_ar_ajc_uf |
| 43 | $D_{4}$ | \( ( 1 + 3 T - 13 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.g_ar_gy_gwn |
| 47 | $C_2^2:C_4$ | \( 1 + 8 T + 17 T^{2} + 380 T^{3} + 4721 T^{4} + 380 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.i_r_oq_gzp |
| 53 | $C_2^2:C_4$ | \( 1 - 4 T + 43 T^{2} - 380 T^{3} + 4761 T^{4} - 380 p T^{5} + 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ae_br_aoq_hbd |
| 59 | $C_2^2:C_4$ | \( 1 + 5 T + T^{2} - 335 T^{3} - 1164 T^{4} - 335 p T^{5} + p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.f_b_amx_absu |
| 61 | $C_2^2:C_4$ | \( 1 + 8 T + 3 T^{2} + 436 T^{3} + 6905 T^{4} + 436 p T^{5} + 3 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.i_d_qu_kfp |
| 67 | $D_{4}$ | \( ( 1 + 11 T + 133 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.w_ox_gng_cllh |
| 71 | $C_2^2:C_4$ | \( 1 - 8 T - 47 T^{2} + 434 T^{3} + 1365 T^{4} + 434 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ai_abv_qs_can |
| 73 | $C_2^2:C_4$ | \( 1 + 14 T + 63 T^{2} - 500 T^{3} - 10579 T^{4} - 500 p T^{5} + 63 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.o_cl_atg_apqx |
| 79 | $C_4\times C_2$ | \( 1 + 30 T + 461 T^{2} + 5400 T^{3} + 52861 T^{4} + 5400 p T^{5} + 461 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.be_rt_hzs_dafd |
| 83 | $C_2^2:C_4$ | \( 1 - 19 T + 103 T^{2} - 695 T^{3} + 10536 T^{4} - 695 p T^{5} + 103 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.at_dz_abat_ppg |
| 89 | $D_{4}$ | \( ( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.k_mt_dpc_cmqv |
| 97 | $C_2^2:C_4$ | \( 1 + 3 T + 47 T^{2} - 255 T^{3} + 3496 T^{4} - 255 p T^{5} + 47 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.d_bv_ajv_fem |
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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
−7.62813128057894544575009415172, −7.24274585264300553559446544602, −7.03209896042145827101206983608, −6.95837542391111750612918221032, −6.76176746997304523722834649025, −6.20689766058446022955717906163, −6.01783340171398422608126946213, −5.86895249492109575255936022108, −5.78439120381694496281224601159, −5.48923909753229337424017550657, −4.89148263516586746769231951056, −4.84458163912750841582333067056, −4.71680045007958691934885134808, −4.18386777719344843121591891684, −3.96590754780426852095939235629, −3.72219353183639255393235694747, −3.50851969793966125428794775709, −3.00350943075617100779362905047, −2.76719870650603692840825878684, −2.49093368070744321759334488814, −2.40515034402199567796537739718, −1.61857486122039439264836990108, −1.52305521024460625325739117740, −1.48015164488613253503274508143, −1.22540484821368605425244983981,
1.22540484821368605425244983981, 1.48015164488613253503274508143, 1.52305521024460625325739117740, 1.61857486122039439264836990108, 2.40515034402199567796537739718, 2.49093368070744321759334488814, 2.76719870650603692840825878684, 3.00350943075617100779362905047, 3.50851969793966125428794775709, 3.72219353183639255393235694747, 3.96590754780426852095939235629, 4.18386777719344843121591891684, 4.71680045007958691934885134808, 4.84458163912750841582333067056, 4.89148263516586746769231951056, 5.48923909753229337424017550657, 5.78439120381694496281224601159, 5.86895249492109575255936022108, 6.01783340171398422608126946213, 6.20689766058446022955717906163, 6.76176746997304523722834649025, 6.95837542391111750612918221032, 7.03209896042145827101206983608, 7.24274585264300553559446544602, 7.62813128057894544575009415172
