| L(s) = 1 | − 6·7-s − 18·13-s − 9·16-s − 9·25-s − 24·31-s − 24·37-s + 18·43-s − 3·49-s − 18·61-s − 64-s − 54·79-s + 108·91-s − 42·103-s + 18·109-s + 54·112-s − 39·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 135·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 4.99·13-s − 9/4·16-s − 9/5·25-s − 4.31·31-s − 3.94·37-s + 2.74·43-s − 3/7·49-s − 2.30·61-s − 1/8·64-s − 6.07·79-s + 11.3·91-s − 4.13·103-s + 1.72·109-s + 5.10·112-s − 3.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 10.3·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + 9 T^{4} + T^{6} + 9 p^{2} T^{8} + p^{6} T^{12} \) | 6.2.a_a_a_j_a_b |
| 5 | \( 1 + 9 T^{2} + 63 T^{4} + 379 T^{6} + 63 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} \) | 6.5.a_j_a_cl_a_op |
| 7 | \( ( 1 + 3 T + 15 T^{2} + 43 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.7.g_bn_gu_bar_dhe_jsp |
| 11 | \( 1 + 39 T^{2} + 843 T^{4} + 11311 T^{6} + 843 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} \) | 6.11.a_bn_a_bgl_a_qtb |
| 13 | \( ( 1 + 9 T + 54 T^{2} + 217 T^{3} + 54 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.13.s_hh_ccc_mek_cfwk_iuvd |
| 17 | \( 1 + 9 T^{2} + 585 T^{4} + 359 p T^{6} + 585 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} \) | 6.17.a_j_a_wn_a_jat |
| 23 | \( 1 + 108 T^{2} + 234 p T^{4} + 157591 T^{6} + 234 p^{3} T^{8} + 108 p^{4} T^{10} + p^{6} T^{12} \) | 6.23.a_ee_a_hza_a_izdf |
| 29 | \( 1 + 81 T^{2} + 4122 T^{4} + 136873 T^{6} + 4122 p^{2} T^{8} + 81 p^{4} T^{10} + p^{6} T^{12} \) | 6.29.a_dd_a_gco_a_humj |
| 31 | \( ( 1 + 12 T + 132 T^{2} + 781 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.31.y_ps_gzy_cnpw_sqkm_elhyd |
| 37 | \( ( 1 + 12 T + 150 T^{2} + 25 p T^{3} + 150 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.37.y_rc_ibq_deoe_zgdw_gppvv |
| 41 | \( 1 + 234 T^{4} + 87211 T^{6} + 234 p^{2} T^{8} + p^{6} T^{12} \) | 6.41.a_a_a_ja_a_ezah |
| 43 | \( ( 1 - 9 T + 144 T^{2} - 773 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.43.as_of_agde_crpa_auxli_gfogv |
| 47 | \( 1 + 99 T^{2} + 8631 T^{4} + 438319 T^{6} + 8631 p^{2} T^{8} + 99 p^{4} T^{10} + p^{6} T^{12} \) | 6.47.a_dv_a_mtz_a_yykl |
| 53 | \( 1 + 198 T^{2} + 18936 T^{4} + 1192555 T^{6} + 18936 p^{2} T^{8} + 198 p^{4} T^{10} + p^{6} T^{12} \) | 6.53.a_hq_a_bcai_a_cpwdn |
| 59 | \( 1 + 153 T^{2} + 17757 T^{4} + 1169119 T^{6} + 17757 p^{2} T^{8} + 153 p^{4} T^{10} + p^{6} T^{12} \) | 6.59.a_fx_a_bagz_a_conmd |
| 61 | \( ( 1 + 9 T + 171 T^{2} + 919 T^{3} + 171 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.61.s_qh_hhc_duph_bgjty_lzdvr |
| 67 | \( ( 1 + 153 T^{2} + 64 T^{3} + 153 p T^{4} + p^{3} T^{6} )^{2} \) | 6.67.a_lu_ey_cmyx_bczg_ieydo |
| 71 | \( 1 + 117 T^{2} + 8154 T^{4} + 550837 T^{6} + 8154 p^{2} T^{8} + 117 p^{4} T^{10} + p^{6} T^{12} \) | 6.71.a_en_a_mbq_a_bfiwb |
| 73 | \( ( 1 + 162 T^{2} + 163 T^{3} + 162 p T^{4} + p^{3} T^{6} )^{2} \) | 6.73.a_mm_mo_cvvc_dadg_kduil |
| 79 | \( ( 1 + 27 T + 432 T^{2} + 4627 T^{3} + 432 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} )^{2} \) | 6.79.cc_cjh_bwfe_bcsra_nnmoc_fdlovx |
| 83 | \( 1 + 291 T^{2} + 42819 T^{4} + 4241671 T^{6} + 42819 p^{2} T^{8} + 291 p^{4} T^{10} + p^{6} T^{12} \) | 6.83.a_lf_a_clix_a_jhirf |
| 89 | \( 1 + 234 T^{2} + 36522 T^{4} + 3597895 T^{6} + 36522 p^{2} T^{8} + 234 p^{4} T^{10} + p^{6} T^{12} \) | 6.89.a_ja_a_ccas_a_hwsip |
| 97 | \( ( 1 + 270 T^{2} - 17 T^{3} + 270 p T^{4} + p^{3} T^{6} )^{2} \) | 6.97.a_uu_abi_hdim_anpc_biynqx |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.00349279389753399905015316823, −4.70836441984577589552931410139, −4.51160413355632658388238937099, −4.42043481429730004906801289533, −4.30549836834073826579599712196, −4.27775010324201701451919195455, −4.16923494180279753381390502175, −3.73833527406634021846464915980, −3.72194063113050042188042672386, −3.58475418366226510381854001377, −3.48321327597282960066412624658, −3.29992716540640132472042889134, −3.06375422195188749304890671335, −2.82746192953369886109053108967, −2.72148466622869908190629663830, −2.70695702883157957233916295922, −2.62929747915078868668548389087, −2.37569648284085426985128880196, −2.03984236180627418681325438061, −2.01097156150050479046043924980, −1.88312738241187430783765964873, −1.77914198108668330559498282868, −1.40328138751594490307185605778, −1.39727102409194435984284841481, −0.885642800704412966476159801711, 0, 0, 0, 0, 0, 0,
0.885642800704412966476159801711, 1.39727102409194435984284841481, 1.40328138751594490307185605778, 1.77914198108668330559498282868, 1.88312738241187430783765964873, 2.01097156150050479046043924980, 2.03984236180627418681325438061, 2.37569648284085426985128880196, 2.62929747915078868668548389087, 2.70695702883157957233916295922, 2.72148466622869908190629663830, 2.82746192953369886109053108967, 3.06375422195188749304890671335, 3.29992716540640132472042889134, 3.48321327597282960066412624658, 3.58475418366226510381854001377, 3.72194063113050042188042672386, 3.73833527406634021846464915980, 4.16923494180279753381390502175, 4.27775010324201701451919195455, 4.30549836834073826579599712196, 4.42043481429730004906801289533, 4.51160413355632658388238937099, 4.70836441984577589552931410139, 5.00349279389753399905015316823
Plot not available for L-functions of degree greater than 10.
