| L(s) = 1 | − 2·2-s + 4·3-s − 4-s − 8·6-s + 6·8-s + 10·9-s − 4·11-s − 4·12-s − 6·16-s − 4·17-s − 20·18-s − 8·19-s + 8·22-s + 24·24-s + 20·27-s − 4·29-s − 8·31-s − 2·32-s − 16·33-s + 8·34-s − 10·36-s − 16·37-s + 16·38-s − 24·41-s − 20·43-s + 4·44-s + 8·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s − 1/2·4-s − 3.26·6-s + 2.12·8-s + 10/3·9-s − 1.20·11-s − 1.15·12-s − 3/2·16-s − 0.970·17-s − 4.71·18-s − 1.83·19-s + 1.70·22-s + 4.89·24-s + 3.84·27-s − 0.742·29-s − 1.43·31-s − 0.353·32-s − 2.78·33-s + 1.37·34-s − 5/3·36-s − 2.63·37-s + 2.59·38-s − 3.74·41-s − 3.04·43-s + 0.603·44-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 5 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| good | 2 | $C_2 \wr C_2\wr C_2$ | \( 1 + p T + 5 T^{2} + 3 p T^{3} + 11 T^{4} + 3 p^{2} T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | 4.2.c_f_g_l |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 36 T^{2} + 124 T^{3} + 557 T^{4} + 124 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.e_bk_eu_vl |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 38 T^{2} - 16 T^{3} + 654 T^{4} - 16 p T^{5} + 38 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_bm_aq_ze |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 58 T^{2} + 188 T^{3} + 1390 T^{4} + 188 p T^{5} + 58 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.e_cg_hg_cbm |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 46 T^{2} + 232 T^{3} + 1222 T^{4} + 232 p T^{5} + 46 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.i_bu_iy_bva |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 60 T^{2} + 20 T^{3} + 1781 T^{4} + 20 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_ci_u_cqn |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 68 T^{2} + 284 T^{3} + 2725 T^{4} + 284 p T^{5} + 68 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.e_cq_ky_eav |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 88 T^{2} + 600 T^{3} + 3986 T^{4} + 600 p T^{5} + 88 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.i_dk_xc_fxi |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 194 T^{2} + 1584 T^{3} + 10995 T^{4} + 1584 p T^{5} + 194 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.q_hm_ciy_qgx |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 348 T^{2} + 3464 T^{3} + 25622 T^{4} + 3464 p T^{5} + 348 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.y_nk_fdg_blxm |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T + 170 T^{2} + 600 T^{3} + 1347 T^{4} + 600 p T^{5} + 170 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.u_go_xc_bzv |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 86 T^{2} - 224 T^{3} + 2638 T^{4} - 224 p T^{5} + 86 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ai_di_aiq_dxm |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 20 T + 304 T^{2} - 3100 T^{3} + 25822 T^{4} - 3100 p T^{5} + 304 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.au_ls_aepg_bmfe |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 158 T^{2} + 1168 T^{3} + 11998 T^{4} + 1168 p T^{5} + 158 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.i_gc_bsy_rtm |
| 61 | $D_{4}$ | \( ( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bg_xo_lie_eacc |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 194 T^{2} + 2008 T^{3} + 18507 T^{4} + 2008 p T^{5} + 194 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.m_hm_czg_bbjv |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 116 T^{2} - 348 T^{3} + 9773 T^{4} - 348 p T^{5} + 116 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ae_em_ank_olx |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 134 T^{2} + 160 T^{3} + 11790 T^{4} + 160 p T^{5} + 134 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_fe_ge_rlm |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 114 T^{2} + 7539 T^{4} + 114 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_ek_a_ldz |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T + 292 T^{2} + 3004 T^{3} + 28002 T^{4} + 3004 p T^{5} + 292 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.u_lg_elo_bpla |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 142 T^{2} - 368 T^{3} + 2742 T^{4} - 368 p T^{5} + 142 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.i_fm_aoe_ebm |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 24 T + 496 T^{2} - 6296 T^{3} + 73730 T^{4} - 6296 p T^{5} + 496 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ay_tc_ajie_efbu |
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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
−6.54473481022027920828883299580, −6.47693593744672993488662634126, −6.04472503493824109352941424297, −5.84481671533476555926171428931, −5.58713382877263590137517566069, −5.21503174759984158074786150512, −5.12397465568752355204684759744, −5.06436279462978827791827758256, −4.86624363016189718558637413110, −4.42528238133269536958100439067, −4.35933305352351622213161281540, −4.19740071203780809956352474056, −4.05927955866094462544364812389, −3.61914152149398555896248875968, −3.46712367133285476673402835798, −3.30631373161888162748513968031, −3.26585353354353028210891026916, −2.75521154273470316594977887855, −2.63337179105892603269424302870, −2.21969832473010720471636952696, −2.03128977493721746063723340253, −2.00496061100021269885579076711, −1.46415938662427907526670372215, −1.39044722992540635999918764078, −1.35342311551794166379461870858, 0, 0, 0, 0,
1.35342311551794166379461870858, 1.39044722992540635999918764078, 1.46415938662427907526670372215, 2.00496061100021269885579076711, 2.03128977493721746063723340253, 2.21969832473010720471636952696, 2.63337179105892603269424302870, 2.75521154273470316594977887855, 3.26585353354353028210891026916, 3.30631373161888162748513968031, 3.46712367133285476673402835798, 3.61914152149398555896248875968, 4.05927955866094462544364812389, 4.19740071203780809956352474056, 4.35933305352351622213161281540, 4.42528238133269536958100439067, 4.86624363016189718558637413110, 5.06436279462978827791827758256, 5.12397465568752355204684759744, 5.21503174759984158074786150512, 5.58713382877263590137517566069, 5.84481671533476555926171428931, 6.04472503493824109352941424297, 6.47693593744672993488662634126, 6.54473481022027920828883299580
