| L(s) = 1 | + 3·2-s + 6·4-s − 3·5-s − 6·7-s + 10·8-s − 3·9-s − 9·10-s − 3·11-s − 18·14-s + 15·16-s − 9·17-s − 9·18-s − 18·20-s − 9·22-s − 15·23-s + 3·25-s − 3·27-s − 36·28-s − 6·29-s + 3·31-s + 21·32-s − 27·34-s + 18·35-s − 18·36-s + 6·37-s − 30·40-s + 9·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s − 1.34·5-s − 2.26·7-s + 3.53·8-s − 9-s − 2.84·10-s − 0.904·11-s − 4.81·14-s + 15/4·16-s − 2.18·17-s − 2.12·18-s − 4.02·20-s − 1.91·22-s − 3.12·23-s + 3/5·25-s − 0.577·27-s − 6.80·28-s − 1.11·29-s + 0.538·31-s + 3.71·32-s − 4.63·34-s + 3.04·35-s − 3·36-s + 0.986·37-s − 4.74·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 19 | | \( 1 \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) | 3.3.a_d_d |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} + 12 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.d_g_m |
| 7 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} + 27 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.g_p_bb |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \) | 3.13.a_v_bd |
| 17 | $S_4\times C_2$ | \( 1 + 9 T + 69 T^{2} + 302 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.j_cr_lq |
| 23 | $S_4\times C_2$ | \( 1 + 15 T + 135 T^{2} + 766 T^{3} + 135 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.p_ff_bdm |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 63 T^{2} + 201 T^{3} + 63 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.g_cl_ht |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} - 52 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_bq_aca |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 87 T^{2} - 348 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_dj_ank |
| 41 | $S_4\times C_2$ | \( 1 - 9 T + 78 T^{2} - 616 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aj_da_axs |
| 43 | $S_4\times C_2$ | \( 1 + 21 T + 222 T^{2} + 1622 T^{3} + 222 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.v_io_ckk |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 81 T^{2} + 456 T^{3} + 81 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.m_dd_ro |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 69 T^{2} - 502 T^{3} + 69 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_cr_ati |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 153 T^{2} - 290 T^{3} + 153 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ad_fx_ale |
| 61 | $S_4\times C_2$ | \( 1 - 24 T + 339 T^{2} - 3120 T^{3} + 339 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ay_nb_aeqa |
| 67 | $S_4\times C_2$ | \( 1 + 105 T^{2} - 123 T^{3} + 105 p T^{4} + p^{3} T^{6} \) | 3.67.a_eb_aet |
| 71 | $S_4\times C_2$ | \( 1 + 21 T + 216 T^{2} + 1676 T^{3} + 216 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.v_ii_cmm |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 213 T^{2} + 426 T^{3} + 213 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.d_if_qk |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 4 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ag_eb_ae |
| 83 | $S_4\times C_2$ | \( 1 + 33 T + 594 T^{2} + 6582 T^{3} + 594 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.bh_ww_jte |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 231 T^{2} + 924 T^{3} + 231 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_ix_bjo |
| 97 | $S_4\times C_2$ | \( 1 + 12 T + 195 T^{2} + 2072 T^{3} + 195 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.m_hn_dbs |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.10427373143592505831041034443, −6.99706658707384228813755959539, −6.85176080252886851655512535261, −6.51167339999349259858063176736, −6.08944499686154071586968192250, −6.08832388813093368293302570029, −5.83979656567785098469937973787, −5.70621425046482971304270388892, −5.61347533713577982534564988568, −5.12338893293399598085920346239, −4.78140971530590111503041636912, −4.48247350217214200766550216927, −4.46367908451319311769642490191, −4.03381794616558863607316665784, −4.00431191848904233915299425427, −3.60399909123137019320919707806, −3.53724244171879380250127942876, −3.13967765210225317753310118247, −3.09964016952368057721082885766, −2.52798494532862335140905539990, −2.46488125006679111133578850310, −2.27886633123105940329912575583, −1.87178490479518050358122177714, −1.55916907006497981232321555379, −0.73997300010006488405696964362, 0, 0, 0,
0.73997300010006488405696964362, 1.55916907006497981232321555379, 1.87178490479518050358122177714, 2.27886633123105940329912575583, 2.46488125006679111133578850310, 2.52798494532862335140905539990, 3.09964016952368057721082885766, 3.13967765210225317753310118247, 3.53724244171879380250127942876, 3.60399909123137019320919707806, 4.00431191848904233915299425427, 4.03381794616558863607316665784, 4.46367908451319311769642490191, 4.48247350217214200766550216927, 4.78140971530590111503041636912, 5.12338893293399598085920346239, 5.61347533713577982534564988568, 5.70621425046482971304270388892, 5.83979656567785098469937973787, 6.08832388813093368293302570029, 6.08944499686154071586968192250, 6.51167339999349259858063176736, 6.85176080252886851655512535261, 6.99706658707384228813755959539, 7.10427373143592505831041034443
