| L(s) = 1 | − 2·3-s − 6·7-s + 9-s + 6·11-s − 4·17-s − 6·19-s + 12·21-s − 6·23-s + 14·25-s + 2·27-s − 2·29-s − 12·33-s + 12·37-s − 14·43-s + 8·49-s + 8·51-s − 4·53-s + 12·57-s + 16·61-s − 6·63-s − 6·67-s + 12·69-s + 18·71-s − 28·75-s − 36·77-s − 8·79-s − 4·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.26·7-s + 1/3·9-s + 1.80·11-s − 0.970·17-s − 1.37·19-s + 2.61·21-s − 1.25·23-s + 14/5·25-s + 0.384·27-s − 0.371·29-s − 2.08·33-s + 1.97·37-s − 2.13·43-s + 8/7·49-s + 1.12·51-s − 0.549·53-s + 1.58·57-s + 2.04·61-s − 0.755·63-s − 0.733·67-s + 1.44·69-s + 2.13·71-s − 3.23·75-s − 4.10·77-s − 0.900·79-s − 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6415022565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6415022565\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ao_a_dv |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.g_bc_ds_lf |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 144 T^{3} + 587 T^{4} - 144 p T^{5} + 36 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ag_bk_afo_wp |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - 19 T^{2} + 4 T^{3} + 664 T^{4} + 4 p T^{5} - 19 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_at_e_zo |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.g_bs_hk_bih |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} - 108 T^{3} - 573 T^{4} - 108 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.g_i_aee_awb |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 7 T^{2} - 94 T^{3} - 836 T^{4} - 94 p T^{5} - 7 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.c_ah_adq_abge |
| 31 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_ado_a_fry |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 109 T^{2} - 732 T^{3} + 4128 T^{4} - 732 p T^{5} + 109 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.am_ef_abce_gcu |
| 41 | $C_2^3$ | \( 1 + 57 T^{2} + 1568 T^{4} + 57 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_cf_a_cii |
| 43 | $D_4\times C_2$ | \( 1 + 14 T + 88 T^{2} + 308 T^{3} + 1387 T^{4} + 308 p T^{5} + 88 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.o_dk_lw_cbj |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_afc_a_mny |
| 53 | $D_{4}$ | \( ( 1 + 2 T + 59 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.e_es_rg_och |
| 59 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_eo_a_plr |
| 61 | $D_4\times C_2$ | \( 1 - 16 T + 73 T^{2} - 16 p T^{3} + 232 p T^{4} - 16 p^{2} T^{5} + 73 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aq_cv_ablo_uyi |
| 67 | $D_4\times C_2$ | \( 1 + 6 T + 100 T^{2} + 528 T^{3} + 4059 T^{4} + 528 p T^{5} + 100 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.g_dw_ui_gad |
| 71 | $D_4\times C_2$ | \( 1 - 18 T + 268 T^{2} - 2880 T^{3} + 28227 T^{4} - 2880 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.as_ki_aegu_bptr |
| 73 | $D_4\times C_2$ | \( 1 - 206 T^{2} + 19539 T^{4} - 206 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_ahy_a_bcxn |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.i_eu_boy_banm |
| 83 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 842 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_adg_a_bgk |
| 89 | $D_4\times C_2$ | \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.am_hu_actc_bdrv |
| 97 | $C_2^3$ | \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_dq_a_awb |
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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
−7.47022779460204591282012657828, −7.30014986394718566548666685583, −6.80456697109641858817930634443, −6.79882523267925290462226280488, −6.58542307033876077635917615890, −6.50421675872121640982230222283, −6.33705870590852585922086579833, −6.21512284159051640079875331244, −5.83124429729933454764378683969, −5.43135843505391145629307869532, −5.38593488460243211188253156563, −4.93664093976204519463266761425, −4.64907809018882189331039822381, −4.43993941734565334596465131686, −4.04011836736178152068722096198, −4.02903381040984637362909945155, −3.47856226575362933397146902544, −3.43578924738439697636436439670, −2.93439916655684820142279672290, −2.72006222092002416178051690180, −2.36185824693527123645621071436, −1.83210314790305442979911372314, −1.46291931553394222282241860079, −0.792849368633957435243971610740, −0.34669280149948333190026072289,
0.34669280149948333190026072289, 0.792849368633957435243971610740, 1.46291931553394222282241860079, 1.83210314790305442979911372314, 2.36185824693527123645621071436, 2.72006222092002416178051690180, 2.93439916655684820142279672290, 3.43578924738439697636436439670, 3.47856226575362933397146902544, 4.02903381040984637362909945155, 4.04011836736178152068722096198, 4.43993941734565334596465131686, 4.64907809018882189331039822381, 4.93664093976204519463266761425, 5.38593488460243211188253156563, 5.43135843505391145629307869532, 5.83124429729933454764378683969, 6.21512284159051640079875331244, 6.33705870590852585922086579833, 6.50421675872121640982230222283, 6.58542307033876077635917615890, 6.79882523267925290462226280488, 6.80456697109641858817930634443, 7.30014986394718566548666685583, 7.47022779460204591282012657828
