| L(s) = 1 | + 3·5-s − 3·7-s − 6·11-s − 3·13-s − 3·19-s − 6·23-s + 15·29-s + 3·31-s − 9·35-s − 3·37-s + 6·41-s − 3·43-s − 15·47-s + 6·49-s + 18·53-s − 18·55-s − 3·59-s − 6·61-s − 9·65-s + 6·67-s − 15·71-s + 9·73-s + 18·77-s − 3·79-s − 18·83-s − 6·89-s + 9·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s − 1.80·11-s − 0.832·13-s − 0.688·19-s − 1.25·23-s + 2.78·29-s + 0.538·31-s − 1.52·35-s − 0.493·37-s + 0.937·41-s − 0.457·43-s − 2.18·47-s + 6/7·49-s + 2.47·53-s − 2.42·55-s − 0.390·59-s − 0.768·61-s − 1.11·65-s + 0.733·67-s − 1.78·71-s + 1.05·73-s + 2.05·77-s − 0.337·79-s − 1.97·83-s − 0.635·89-s + 0.943·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 3 T + 9 T^{2} - 21 T^{3} + 9 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ad_j_av |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 123 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.g_bk_et |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) | 3.13.d_bq_db |
| 17 | $S_4\times C_2$ | \( 1 + 18 T^{2} - 9 T^{3} + 18 p T^{4} + p^{3} T^{6} \) | 3.17.a_s_aj |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 65 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.d_v_cn |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 54 T^{2} + 177 T^{3} + 54 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_cc_gv |
| 29 | $S_4\times C_2$ | \( 1 - 15 T + 135 T^{2} - 807 T^{3} + 135 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ap_ff_abfb |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 133 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_p_afd |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 115 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.d_bh_el |
| 41 | $S_4\times C_2$ | \( 1 - 6 T + 126 T^{2} - 483 T^{3} + 126 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_ew_asp |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 21 T^{2} - 205 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.d_v_ahx |
| 47 | $S_4\times C_2$ | \( 1 + 15 T + 177 T^{2} + 1329 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.p_gv_bzd |
| 53 | $S_4\times C_2$ | \( 1 - 18 T + 198 T^{2} - 1521 T^{3} + 198 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.as_hq_acgn |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 435 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.d_et_qt |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 138 T^{2} + 763 T^{3} + 138 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.g_fi_bdj |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 174 T^{2} - 745 T^{3} + 174 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_gs_abcr |
| 71 | $S_4\times C_2$ | \( 1 + 15 T + 231 T^{2} + 1833 T^{3} + 231 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.p_ix_csn |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1235 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_hz_abvn |
| 79 | $S_4\times C_2$ | \( 1 + 3 T + 3 T^{2} - 889 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_d_abif |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 324 T^{2} + 3015 T^{3} + 324 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.s_mm_elz |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 72 T^{2} - 21 T^{3} + 72 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_cu_av |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 309 T^{2} + 2887 T^{3} + 309 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.p_lx_ehb |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.08246219336223730278599909958, −6.74932178426159151211503723077, −6.74190547808909234460128251145, −6.58586361181628938557659854063, −6.16323292788203394463054294791, −6.04775979602363786613911236998, −5.80888869093487151167175480274, −5.58504680372247767555354092820, −5.33691546114439179531117017264, −5.24190087147731345824048624971, −4.79903003124735582742140500034, −4.58456609371435421357003845192, −4.58426118328779961310628135331, −3.88739525688595846308368500953, −3.86179601999682885406767069923, −3.82509735919707138344176673540, −2.92696143491461422645470771017, −2.89596425462122661069609151749, −2.86155397113983188254064112432, −2.53193404716971418050283492645, −2.29318559644457245182876246721, −2.08239980023593701816743745846, −1.54517886170177523460438817871, −1.26290115106294334373966191833, −1.00212846262673386022397324323, 0, 0, 0,
1.00212846262673386022397324323, 1.26290115106294334373966191833, 1.54517886170177523460438817871, 2.08239980023593701816743745846, 2.29318559644457245182876246721, 2.53193404716971418050283492645, 2.86155397113983188254064112432, 2.89596425462122661069609151749, 2.92696143491461422645470771017, 3.82509735919707138344176673540, 3.86179601999682885406767069923, 3.88739525688595846308368500953, 4.58426118328779961310628135331, 4.58456609371435421357003845192, 4.79903003124735582742140500034, 5.24190087147731345824048624971, 5.33691546114439179531117017264, 5.58504680372247767555354092820, 5.80888869093487151167175480274, 6.04775979602363786613911236998, 6.16323292788203394463054294791, 6.58586361181628938557659854063, 6.74190547808909234460128251145, 6.74932178426159151211503723077, 7.08246219336223730278599909958
