| L(s) = 1 | + 2-s + 4-s + 4·5-s + 8·7-s + 8-s + 4·10-s + 15·13-s + 8·14-s − 2·16-s − 6·17-s + 19-s + 4·20-s + 23-s + 10·25-s + 15·26-s + 8·28-s + 17·29-s + 15·31-s − 5·32-s − 6·34-s + 32·35-s − 37-s + 38-s + 4·40-s − 12·41-s + 14·43-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 3.02·7-s + 0.353·8-s + 1.26·10-s + 4.16·13-s + 2.13·14-s − 1/2·16-s − 1.45·17-s + 0.229·19-s + 0.894·20-s + 0.208·23-s + 2·25-s + 2.94·26-s + 1.51·28-s + 3.15·29-s + 2.69·31-s − 0.883·32-s − 1.02·34-s + 5.40·35-s − 0.164·37-s + 0.162·38-s + 0.632·40-s − 1.87·41-s + 2.13·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(67.70861053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(67.70861053\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 11 | | \( 1 \) | |
| good | 2 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 3 T^{4} - p^{3} T^{7} + p^{4} T^{8} \) | 4.2.ab_a_a_d |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 36 T^{2} - 111 T^{3} + 307 T^{4} - 111 p T^{5} + 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.7.ai_bk_aeh_lv |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 128 T^{2} - 735 T^{3} + 3089 T^{4} - 735 p T^{5} + 128 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ap_ey_abch_eov |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 48 T^{2} + 12 p T^{3} + 1153 T^{4} + 12 p^{2} T^{5} + 48 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.g_bw_hw_bsj |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 66 T^{2} - 45 T^{3} + 1795 T^{4} - 45 p T^{5} + 66 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ab_co_abt_crb |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 53 T^{2} - 40 T^{3} + 1721 T^{4} - 40 p T^{5} + 53 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ab_cb_abo_cof |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 17 T + 164 T^{2} - 1211 T^{3} + 7211 T^{4} - 1211 p T^{5} + 164 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ar_gi_abup_krj |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 159 T^{2} - 1270 T^{3} + 7911 T^{4} - 1270 p T^{5} + 159 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ap_gd_abww_lsh |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + T + 65 T^{2} + 90 T^{3} + 3223 T^{4} + 90 p T^{5} + 65 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.37.b_cn_dm_etz |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 202 T^{2} + 1513 T^{3} + 13213 T^{4} + 1513 p T^{5} + 202 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.m_hu_cgf_tof |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 14 T + 222 T^{2} - 1863 T^{3} + 15403 T^{4} - 1863 p T^{5} + 222 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ao_io_actr_wul |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 14 T + 154 T^{2} + 19 p T^{3} + 6549 T^{4} + 19 p^{2} T^{5} + 154 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.o_fy_bij_jrx |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 130 T^{2} + 235 T^{3} + 9873 T^{4} + 235 p T^{5} + 130 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.c_fa_jb_opt |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 237 T^{2} - 1758 T^{3} + 20905 T^{4} - 1758 p T^{5} + 237 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.al_jd_acpq_beyb |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + T + 156 T^{2} + 77 T^{3} + 12701 T^{4} + 77 p T^{5} + 156 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) | 4.61.b_ga_cz_sun |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 194 T^{2} - 825 T^{3} + 18067 T^{4} - 825 p T^{5} + 194 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.af_hm_abft_basx |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T + 139 T^{2} + 60 T^{3} + 8835 T^{4} + 60 p T^{5} + 139 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ad_fj_ci_nbv |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 45 T + 966 T^{2} - 13205 T^{3} + 130227 T^{4} - 13205 p T^{5} + 966 p^{2} T^{6} - 45 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.abt_ble_atnx_hkqt |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 87 T^{2} - 970 T^{3} + 423 T^{4} - 970 p T^{5} + 87 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_dj_abli_qh |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 337 T^{2} - 3110 T^{3} + 40189 T^{4} - 3110 p T^{5} + 337 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ap_mz_aepq_chlt |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 301 T^{2} + 378 T^{3} + 37835 T^{4} + 378 p T^{5} + 301 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.c_lp_oo_cdzf |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 26 T + 620 T^{2} + 8385 T^{3} + 102353 T^{4} + 8385 p T^{5} + 620 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ba_xw_mkn_fvkr |
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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
−5.76176239762648355745361511264, −5.39182330356323177750164078560, −5.26432820402692645652365818027, −5.25822741610642828186061204667, −5.02112796466570766490693328111, −4.72941809648093611762136090467, −4.67507080585574103331711072516, −4.44092682017376867618305736664, −4.39355285650658825430265200687, −4.05492102231944503142328282277, −3.76816832860453153938117962940, −3.74056132627327208091897947922, −3.40589634315995502727045823901, −3.06038572869903131883552616438, −2.97781978099480904603460561385, −2.65493482888930082093833472926, −2.29235822945112451751759842913, −2.28138863172614083010601223482, −2.03592535838504747184062087675, −1.59518688374114361141660098075, −1.56308575284086032733971288412, −1.45285941210564710839932435333, −0.895259090842825264976762090486, −0.861523933421678586911279070750, −0.800767071220829069745730041736,
0.800767071220829069745730041736, 0.861523933421678586911279070750, 0.895259090842825264976762090486, 1.45285941210564710839932435333, 1.56308575284086032733971288412, 1.59518688374114361141660098075, 2.03592535838504747184062087675, 2.28138863172614083010601223482, 2.29235822945112451751759842913, 2.65493482888930082093833472926, 2.97781978099480904603460561385, 3.06038572869903131883552616438, 3.40589634315995502727045823901, 3.74056132627327208091897947922, 3.76816832860453153938117962940, 4.05492102231944503142328282277, 4.39355285650658825430265200687, 4.44092682017376867618305736664, 4.67507080585574103331711072516, 4.72941809648093611762136090467, 5.02112796466570766490693328111, 5.25822741610642828186061204667, 5.26432820402692645652365818027, 5.39182330356323177750164078560, 5.76176239762648355745361511264
