| L(s) = 1 | + 4·2-s + 10·4-s + 4·5-s + 4·7-s + 20·8-s − 2·9-s + 16·10-s − 3·11-s − 8·13-s + 16·14-s + 35·16-s − 5·17-s − 8·18-s + 3·19-s + 40·20-s − 12·22-s − 32·26-s + 27-s + 40·28-s + 9·29-s − 4·31-s + 56·32-s − 20·34-s + 16·35-s − 20·36-s + 4·37-s + 12·38-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 1.78·5-s + 1.51·7-s + 7.07·8-s − 2/3·9-s + 5.05·10-s − 0.904·11-s − 2.21·13-s + 4.27·14-s + 35/4·16-s − 1.21·17-s − 1.88·18-s + 0.688·19-s + 8.94·20-s − 2.55·22-s − 6.27·26-s + 0.192·27-s + 7.55·28-s + 1.67·29-s − 0.718·31-s + 9.89·32-s − 3.42·34-s + 2.70·35-s − 3.33·36-s + 0.657·37-s + 1.94·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(79.28574393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(79.28574393\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 23 | | \( 1 \) | |
| good | 3 | $C_2 \wr S_4$ | \( 1 + 2 T^{2} - T^{3} - 2 T^{4} - p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_c_ab_ac |
| 5 | $C_2 \wr S_4$ | \( 1 - 4 T + 16 T^{2} - 9 p T^{3} + 106 T^{4} - 9 p^{2} T^{5} + 16 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ae_q_abt_ec |
| 11 | $C_2 \wr S_4$ | \( 1 + 3 T + 13 T^{2} + 31 T^{3} + 108 T^{4} + 31 p T^{5} + 13 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.d_n_bf_ee |
| 13 | $C_2 \wr S_4$ | \( 1 + 8 T + 34 T^{2} + 177 T^{3} + 64 p T^{4} + 177 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.i_bi_gv_bga |
| 17 | $C_2 \wr S_4$ | \( 1 + 5 T + 29 T^{2} + 142 T^{3} + 906 T^{4} + 142 p T^{5} + 29 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.f_bd_fm_biw |
| 19 | $C_2 \wr S_4$ | \( 1 - 3 T + 39 T^{2} - 92 T^{3} + 784 T^{4} - 92 p T^{5} + 39 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ad_bn_ado_bee |
| 29 | $C_2 \wr S_4$ | \( 1 - 9 T + 98 T^{2} - 611 T^{3} + 4354 T^{4} - 611 p T^{5} + 98 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.aj_du_axn_glm |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 81 T^{2} + 342 T^{3} + 3372 T^{4} + 342 p T^{5} + 81 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.e_dd_ne_ezs |
| 37 | $C_2 \wr S_4$ | \( 1 - 4 T + 105 T^{2} - 286 T^{3} + 5176 T^{4} - 286 p T^{5} + 105 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ae_eb_ala_hrc |
| 41 | $C_2 \wr S_4$ | \( 1 + 88 T^{2} - 289 T^{3} + 3596 T^{4} - 289 p T^{5} + 88 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_dk_ald_fii |
| 43 | $C_2 \wr S_4$ | \( 1 - 13 T + 123 T^{2} - 1084 T^{3} + 8956 T^{4} - 1084 p T^{5} + 123 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.an_et_abps_ngm |
| 47 | $C_2 \wr S_4$ | \( 1 + T + 111 T^{2} - 199 T^{3} + 5632 T^{4} - 199 p T^{5} + 111 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.47.b_eh_ahr_iiq |
| 53 | $C_2 \wr S_4$ | \( 1 - 17 T + 272 T^{2} - 2599 T^{3} + 23238 T^{4} - 2599 p T^{5} + 272 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ar_km_advz_biju |
| 59 | $C_2 \wr S_4$ | \( 1 + 7 T + 170 T^{2} + 699 T^{3} + 12234 T^{4} + 699 p T^{5} + 170 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.h_go_bax_sco |
| 61 | $C_2 \wr S_4$ | \( 1 - 33 T + 604 T^{2} - 7487 T^{3} + 68158 T^{4} - 7487 p T^{5} + 604 p^{2} T^{6} - 33 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.abh_xg_albz_dwvm |
| 67 | $C_2 \wr S_4$ | \( 1 - T + 200 T^{2} + 71 T^{3} + 17566 T^{4} + 71 p T^{5} + 200 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ab_hs_ct_zzq |
| 71 | $C_2 \wr S_4$ | \( 1 + 35 T + 677 T^{2} + 8961 T^{3} + 87628 T^{4} + 8961 p T^{5} + 677 p^{2} T^{6} + 35 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.bj_bab_ngr_ezqi |
| 73 | $C_2 \wr S_4$ | \( 1 + 10 T + 168 T^{2} + 1284 T^{3} + 13481 T^{4} + 1284 p T^{5} + 168 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.k_gm_bxk_tyn |
| 79 | $C_2 \wr S_4$ | \( 1 - 12 T + 247 T^{2} - 1544 T^{3} + 22576 T^{4} - 1544 p T^{5} + 247 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.am_jn_achk_bhki |
| 83 | $C_2 \wr S_4$ | \( 1 - 12 T + 344 T^{2} - 2843 T^{3} + 43462 T^{4} - 2843 p T^{5} + 344 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.am_ng_aefj_cmhq |
| 89 | $C_2 \wr S_4$ | \( 1 - 3 T + 253 T^{2} - 850 T^{3} + 29378 T^{4} - 850 p T^{5} + 253 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.ad_jt_abgs_brly |
| 97 | $C_2 \wr S_4$ | \( 1 + 5 T + 9 T^{2} - 258 T^{3} + 8406 T^{4} - 258 p T^{5} + 9 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.f_j_ajy_mli |
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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
−5.52225332582941351497866931948, −5.20062220781534881562842184274, −5.17315552305538698254442231653, −5.12149297377160277029584641070, −5.03780839438364401040476495533, −4.55605466392318645901764905221, −4.36678918712879736214243350070, −4.35203204799476122942156352325, −4.15707845701062635699903905324, −4.08991664442691068272134232848, −3.80396474582986056838115681877, −3.37772009050970962399322605861, −3.21975577067821100708177520254, −2.92559924945595629592978060692, −2.75594890235497438980208744384, −2.61326620860946065611090365628, −2.49628095708376626540982444806, −2.18637390746383652568657865976, −2.16585119081641227477853795034, −1.95208074512707627757797136947, −1.70747260419514887331293817446, −1.47530821501562413943378781004, −0.993044951670342295043036433692, −0.72397068007173777457049751413, −0.41398387249139837181614916698,
0.41398387249139837181614916698, 0.72397068007173777457049751413, 0.993044951670342295043036433692, 1.47530821501562413943378781004, 1.70747260419514887331293817446, 1.95208074512707627757797136947, 2.16585119081641227477853795034, 2.18637390746383652568657865976, 2.49628095708376626540982444806, 2.61326620860946065611090365628, 2.75594890235497438980208744384, 2.92559924945595629592978060692, 3.21975577067821100708177520254, 3.37772009050970962399322605861, 3.80396474582986056838115681877, 4.08991664442691068272134232848, 4.15707845701062635699903905324, 4.35203204799476122942156352325, 4.36678918712879736214243350070, 4.55605466392318645901764905221, 5.03780839438364401040476495533, 5.12149297377160277029584641070, 5.17315552305538698254442231653, 5.20062220781534881562842184274, 5.52225332582941351497866931948
