| 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
| \(L(1)\) |
\(\approx\) |
\(2.583622589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.583622589\) |
| \(L(\frac{3}{2})\) |
|
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.e_d_ak_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.c_ad_au_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.f_v_fp_bku |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ae_dk_aka_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.ac_abb_em_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.ag_ar_agy_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.ai_r_aoq_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.af_b_mx_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.aw_ox_agng_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.i_abv_aqs_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.abe_rt_ahzs_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.t_dz_bat_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
−7.49443635470982177868674808244, −6.95559045343623509276794988917, −6.86593511442014058207956331485, −6.68332659587717983714232691594, −6.48003365121871756651024438426, −6.28581107934889659182686803034, −5.99731451620617575832785918930, −5.95738040785143712591156216789, −5.77377233876305941554149123651, −5.31651335006980100606625237504, −5.19422661765513422226681008222, −5.06960779204514328876710123208, −4.96375545083315788195811752620, −4.27026680959397541433247549472, −4.02059519121880717507133596323, −3.92845744287755153731773295868, −3.90954321319529394783451424457, −2.97683841371023674867091274333, −2.73149174884401650336538326864, −2.44511683272573384922540040360, −2.28737442491466018081406974505, −1.48894032084849305696760035562, −1.39336544754811113656699097141, −1.12734309260774095529494071713, −0.57752535777073462517267143835,
0.57752535777073462517267143835, 1.12734309260774095529494071713, 1.39336544754811113656699097141, 1.48894032084849305696760035562, 2.28737442491466018081406974505, 2.44511683272573384922540040360, 2.73149174884401650336538326864, 2.97683841371023674867091274333, 3.90954321319529394783451424457, 3.92845744287755153731773295868, 4.02059519121880717507133596323, 4.27026680959397541433247549472, 4.96375545083315788195811752620, 5.06960779204514328876710123208, 5.19422661765513422226681008222, 5.31651335006980100606625237504, 5.77377233876305941554149123651, 5.95738040785143712591156216789, 5.99731451620617575832785918930, 6.28581107934889659182686803034, 6.48003365121871756651024438426, 6.68332659587717983714232691594, 6.86593511442014058207956331485, 6.95559045343623509276794988917, 7.49443635470982177868674808244
