| L(s) = 1 | + 4·4-s − 8·5-s + 4·7-s + 6·9-s − 16·13-s + 12·16-s − 8·17-s + 12·19-s − 32·20-s + 8·23-s + 38·25-s + 16·28-s + 8·29-s − 32·35-s + 24·36-s − 4·37-s + 16·43-s − 48·45-s + 8·49-s − 64·52-s + 24·59-s − 8·61-s + 24·63-s + 32·64-s + 128·65-s + 8·67-s − 32·68-s + ⋯ |
| L(s) = 1 | + 2·4-s − 3.57·5-s + 1.51·7-s + 2·9-s − 4.43·13-s + 3·16-s − 1.94·17-s + 2.75·19-s − 7.15·20-s + 1.66·23-s + 38/5·25-s + 3.02·28-s + 1.48·29-s − 5.40·35-s + 4·36-s − 0.657·37-s + 2.43·43-s − 7.15·45-s + 8/7·49-s − 8.87·52-s + 3.12·59-s − 1.02·61-s + 3.02·63-s + 4·64-s + 15.8·65-s + 0.977·67-s − 3.88·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.998225232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.998225232\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) | |
| 11 | $C_2^2$ | \( 1 + T^{4} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_ag_a_t |
| 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.ae_i_a_abx |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.q_fo_bgq_fio |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 192 T^{3} + 1103 T^{4} + 192 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.i_bg_hk_bql |
| 19 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 432 T^{3} + 2303 T^{4} - 432 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.am_cu_aqq_dkp |
| 23 | $C_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 216 T^{3} + 1442 T^{4} - 216 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ai_bg_aii_cdm |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 96 T^{3} - T^{4} - 96 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ai_bg_ads_ab |
| 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 | $D_{4}$ | \( ( 1 + 2 T + 43 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.e_dm_mi_hfv |
| 41 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_afk_a_maw |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.aq_hw_acro_uwk |
| 47 | $C_2^3$ | \( 1 - 1054 T^{4} + p^{4} T^{8} \) | 4.47.a_a_a_aboo |
| 53 | $D_4\times C_2$ | \( 1 - 50 T^{2} - 29 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_aby_a_abd |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 3048 T^{3} + 27634 T^{4} - 3048 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ay_lc_aeng_boww |
| 61 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 160 T^{3} - 881 T^{4} + 160 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.i_bg_ge_abhx |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 106 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.ai_iu_acbg_bhby |
| 71 | $D_{4}$ | \( ( 1 + 14 T + 189 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.bc_wc_kua_eeyd |
| 73 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 136 T^{3} - 2558 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.i_bg_fg_aduk |
| 79 | $D_{4}$ | \( ( 1 - 20 T + 250 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.abo_biq_atme_hwks |
| 83 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 29874 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ajw_a_bsfa |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.abk_bdy_aqvo_hczj |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 2860 T^{3} + 38782 T^{4} + 2860 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.u_hs_ega_cfjq |
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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
−7.24313306503659371598753238999, −7.22750151619839113866231707875, −7.08362480488839292891714437841, −6.97353591316705511061082510204, −6.76520259756054669552671845313, −6.14921602363705541762862207209, −6.04706002085833651059607217795, −5.60235462515091764519257478328, −5.00315098928567643981209219658, −4.92609442915976270290500327857, −4.84912490077176874699204809371, −4.78718960485893909347518981986, −4.46081470516382565398233875428, −4.46050444639846946084999424447, −3.82525185109316646379559728340, −3.45031994474868542267005470640, −3.44108298125976714618790609022, −3.19419251357760387205690173265, −2.65017767572626639415171974811, −2.40131266509949980742423495521, −2.19711817168338725602268407854, −2.05681817015448399903905930524, −1.06258331573150795468223195370, −0.990299890629391258340857550567, −0.61021036060242474873276355204,
0.61021036060242474873276355204, 0.990299890629391258340857550567, 1.06258331573150795468223195370, 2.05681817015448399903905930524, 2.19711817168338725602268407854, 2.40131266509949980742423495521, 2.65017767572626639415171974811, 3.19419251357760387205690173265, 3.44108298125976714618790609022, 3.45031994474868542267005470640, 3.82525185109316646379559728340, 4.46050444639846946084999424447, 4.46081470516382565398233875428, 4.78718960485893909347518981986, 4.84912490077176874699204809371, 4.92609442915976270290500327857, 5.00315098928567643981209219658, 5.60235462515091764519257478328, 6.04706002085833651059607217795, 6.14921602363705541762862207209, 6.76520259756054669552671845313, 6.97353591316705511061082510204, 7.08362480488839292891714437841, 7.22750151619839113866231707875, 7.24313306503659371598753238999
