| L(s) = 1 | − 3·5-s − 3·7-s − 3·13-s + 6·17-s + 9·19-s − 6·23-s − 6·25-s − 3·29-s + 9·31-s + 9·35-s − 15·37-s + 6·41-s + 15·43-s − 9·47-s + 6·49-s − 6·53-s − 3·59-s − 12·61-s + 9·65-s + 12·67-s − 21·71-s + 3·73-s + 15·79-s + 6·83-s − 18·85-s − 18·89-s + 9·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s − 0.832·13-s + 1.45·17-s + 2.06·19-s − 1.25·23-s − 6/5·25-s − 0.557·29-s + 1.61·31-s + 1.52·35-s − 2.46·37-s + 0.937·41-s + 2.28·43-s − 1.31·47-s + 6/7·49-s − 0.824·53-s − 0.390·59-s − 1.53·61-s + 1.11·65-s + 1.46·67-s − 2.49·71-s + 0.351·73-s + 1.68·79-s + 0.658·83-s − 1.95·85-s − 1.90·89-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 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 29 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.d_p_bd |
| 11 | $A_4\times C_2$ | \( 1 + 24 T^{2} + 9 T^{3} + 24 p T^{4} + p^{3} T^{6} \) | 3.11.a_y_j |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 30 T^{2} + 75 T^{3} + 30 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.d_be_cx |
| 17 | $A_4\times C_2$ | \( 1 - 6 T + 54 T^{2} - 203 T^{3} + 54 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ag_cc_ahv |
| 19 | $A_4\times C_2$ | \( 1 - 9 T + 63 T^{2} - 17 p T^{3} + 63 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.aj_cl_aml |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 24 T^{2} + 7 T^{3} + 24 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_y_h |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 155 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.d_bz_fz |
| 31 | $A_4\times C_2$ | \( 1 - 9 T + 99 T^{2} - 485 T^{3} + 99 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.aj_dv_asr |
| 37 | $A_4\times C_2$ | \( 1 + 15 T + 165 T^{2} + 1167 T^{3} + 165 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.p_gj_bsx |
| 41 | $A_4\times C_2$ | \( 1 - 6 T + 108 T^{2} - 473 T^{3} + 108 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_ee_asf |
| 43 | $A_4\times C_2$ | \( 1 - 15 T + 183 T^{2} - 1273 T^{3} + 183 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ap_hb_abwz |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 147 T^{2} + 847 T^{3} + 147 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.j_fr_bgp |
| 53 | $A_4\times C_2$ | \( 1 + 6 T + 162 T^{2} + 617 T^{3} + 162 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_gg_xt |
| 59 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} - 135 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.d_bh_aff |
| 61 | $A_4\times C_2$ | \( 1 + 12 T + 192 T^{2} + 1391 T^{3} + 192 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.m_hk_cbn |
| 67 | $A_4\times C_2$ | \( 1 - 12 T + 156 T^{2} - 1281 T^{3} + 156 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.am_ga_abxh |
| 71 | $A_4\times C_2$ | \( 1 + 21 T + 249 T^{2} + 2115 T^{3} + 249 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.v_jp_ddj |
| 73 | $A_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 475 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ad_eb_ash |
| 79 | $A_4\times C_2$ | \( 1 - 15 T + 195 T^{2} - 1433 T^{3} + 195 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ap_hn_acdd |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 78 T^{2} - 1539 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ag_da_achf |
| 89 | $A_4\times C_2$ | \( 1 + 18 T + 318 T^{2} + 3241 T^{3} + 318 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.s_mg_eur |
| 97 | $A_4\times C_2$ | \( 1 - 15 T + 57 T^{2} + 311 T^{3} + 57 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ap_cf_lz |
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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
−7.43836049855929291136177812717, −6.78407863329830134339242124153, −6.77133277742767486719196996223, −6.71497642080389567538508251470, −6.10540295049744445162447605505, −6.01190926414420686246449392595, −5.81600657110481492917716530867, −5.63250505667332133142670622449, −5.26491110432221434367724619595, −5.24562807308584917405946310195, −4.69589698250356102939267367609, −4.55899375642847411583014006628, −4.43031461681743144700814498768, −3.92692586920402492690842809997, −3.77500940206599460789082069310, −3.59428029316419369696975190218, −3.35739813348413944612407835129, −3.11062385148048157066934025733, −3.01432073810221349152292388015, −2.36667682377777927982569163951, −2.36542809622682240104197867845, −2.02380948398508516859410606979, −1.36349556034148910598468215275, −1.12578964215241446131127742739, −1.00932467503246526418867073819, 0, 0, 0,
1.00932467503246526418867073819, 1.12578964215241446131127742739, 1.36349556034148910598468215275, 2.02380948398508516859410606979, 2.36542809622682240104197867845, 2.36667682377777927982569163951, 3.01432073810221349152292388015, 3.11062385148048157066934025733, 3.35739813348413944612407835129, 3.59428029316419369696975190218, 3.77500940206599460789082069310, 3.92692586920402492690842809997, 4.43031461681743144700814498768, 4.55899375642847411583014006628, 4.69589698250356102939267367609, 5.24562807308584917405946310195, 5.26491110432221434367724619595, 5.63250505667332133142670622449, 5.81600657110481492917716530867, 6.01190926414420686246449392595, 6.10540295049744445162447605505, 6.71497642080389567538508251470, 6.77133277742767486719196996223, 6.78407863329830134339242124153, 7.43836049855929291136177812717
