| L(s) = 1 | − 4·2-s + 4·3-s + 10·4-s − 16·6-s − 20·8-s + 10·9-s − 4·11-s + 40·12-s + 35·16-s − 40·18-s + 16·22-s − 8·23-s − 80·24-s − 4·25-s + 20·27-s + 16·31-s − 56·32-s − 16·33-s + 100·36-s + 8·37-s + 8·43-s − 40·44-s + 32·46-s + 16·47-s + 140·48-s + 16·50-s + 8·53-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s − 6.53·6-s − 7.07·8-s + 10/3·9-s − 1.20·11-s + 11.5·12-s + 35/4·16-s − 9.42·18-s + 3.41·22-s − 1.66·23-s − 16.3·24-s − 4/5·25-s + 3.84·27-s + 2.87·31-s − 9.89·32-s − 2.78·33-s + 50/3·36-s + 1.31·37-s + 1.21·43-s − 6.03·44-s + 4.71·46-s + 2.33·47-s + 20.2·48-s + 2.26·50-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.115718272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.115718272\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 7 | | \( 1 \) | |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 4 T^{2} + 46 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_e_a_bu |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T^{2} - 32 T^{3} + 242 T^{4} - 32 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_q_abg_ji |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 36 T^{2} + 32 T^{3} + 638 T^{4} + 32 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_bk_bg_yo |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 40 T^{2} + 80 T^{3} + 794 T^{4} + 80 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_bo_dc_beo |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 36 T^{2} - 56 T^{3} - 570 T^{4} - 56 p T^{5} + 36 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.i_bk_ace_avy |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_dw_a_gew |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 200 T^{2} - 1568 T^{3} + 10378 T^{4} - 1568 p T^{5} + 200 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.aq_hs_acii_pje |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 76 T^{2} - 408 T^{3} + 2486 T^{4} - 408 p T^{5} + 76 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ai_cy_aps_drq |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 132 T^{2} - 32 T^{3} + 7454 T^{4} - 32 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_fc_abg_las |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 116 T^{2} - 424 T^{3} + 5110 T^{4} - 424 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ai_em_aqi_hoo |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 248 T^{2} - 2304 T^{3} + 18890 T^{4} - 2304 p T^{5} + 248 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.aq_jo_adkq_bbyo |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 140 T^{2} - 1048 T^{3} + 9334 T^{4} - 1048 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ai_fk_aboi_nva |
| 59 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.abg_xg_alau_dvuc |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 304 T^{2} - 2864 T^{3} + 29362 T^{4} - 2864 p T^{5} + 304 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aq_ls_aege_brli |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 124 T^{2} - 1104 T^{3} + 11510 T^{4} - 1104 p T^{5} + 124 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.aq_eu_abqm_ras |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 140 T^{2} - 256 T^{3} + 12422 T^{4} - 256 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_fk_ajw_sju |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 116 T^{2} - 448 T^{3} + 6462 T^{4} - 448 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_em_arg_joo |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 236 T^{2} - 2448 T^{3} + 24582 T^{4} - 2448 p T^{5} + 236 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.aq_jc_adqe_bkjm |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 280 T^{2} - 48 T^{3} + 32794 T^{4} - 48 p T^{5} + 280 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ku_abw_bwni |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 160 T^{2} - 1904 T^{3} + 24642 T^{4} - 1904 p T^{5} + 160 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.aq_ge_acvg_bklu |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 352 T^{2} + 3984 T^{3} + 51458 T^{4} + 3984 p T^{5} + 352 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.q_no_fxg_cyde |
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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
−6.48276837195326434996856459618, −5.85267106514597354798289223264, −5.79850295649733796048217731174, −5.68653489664922849437200161513, −5.66751349203700436250226642909, −5.04846149200545799243478761073, −4.92871202731708751630146106394, −4.65796789384715268447361979317, −4.41949677115148371712792643919, −3.93618836947834020180445358727, −3.85465032870054956556928133328, −3.77466704932549821531615731445, −3.73620559926979813057613887710, −3.06303823879499339361045703378, −2.91026403855153755392812581941, −2.78472330157732655701162285431, −2.52639349065783285134585201865, −2.22237085752925003982935036300, −2.10516790966525592416541477660, −2.00581310805131738547428938755, −1.98493571374003736049485483123, −1.00170015497670842844860900267, −0.987418545566388354409682701717, −0.65443612567779958448135464163, −0.63489052297660975331385629687,
0.63489052297660975331385629687, 0.65443612567779958448135464163, 0.987418545566388354409682701717, 1.00170015497670842844860900267, 1.98493571374003736049485483123, 2.00581310805131738547428938755, 2.10516790966525592416541477660, 2.22237085752925003982935036300, 2.52639349065783285134585201865, 2.78472330157732655701162285431, 2.91026403855153755392812581941, 3.06303823879499339361045703378, 3.73620559926979813057613887710, 3.77466704932549821531615731445, 3.85465032870054956556928133328, 3.93618836947834020180445358727, 4.41949677115148371712792643919, 4.65796789384715268447361979317, 4.92871202731708751630146106394, 5.04846149200545799243478761073, 5.66751349203700436250226642909, 5.68653489664922849437200161513, 5.79850295649733796048217731174, 5.85267106514597354798289223264, 6.48276837195326434996856459618
