| L(s) = 1 | + 2·5-s − 3·7-s − 3·11-s + 3·17-s − 6·19-s + 23-s − 3·25-s + 13·29-s + 8·31-s − 6·35-s + 11·37-s + 11·41-s − 13·43-s + 47-s + 3·49-s + 8·53-s − 6·55-s + 7·59-s − 12·61-s − 8·67-s + 20·71-s − 4·73-s + 9·77-s − 27·79-s + 16·83-s + 6·85-s + 18·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.13·7-s − 0.904·11-s + 0.727·17-s − 1.37·19-s + 0.208·23-s − 3/5·25-s + 2.41·29-s + 1.43·31-s − 1.01·35-s + 1.80·37-s + 1.71·41-s − 1.98·43-s + 0.145·47-s + 3/7·49-s + 1.09·53-s − 0.809·55-s + 0.911·59-s − 1.53·61-s − 0.977·67-s + 2.37·71-s − 0.468·73-s + 1.02·77-s − 3.03·79-s + 1.75·83-s + 0.650·85-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{9} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.699265333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.699265333\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 8 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ac_h_ai |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} - T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.d_g_ab |
| 13 | $S_4\times C_2$ | \( 1 + 15 T^{2} - 36 T^{3} + 15 p T^{4} + p^{3} T^{6} \) | 3.13.a_p_abk |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} + 15 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ad_s_p |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 8 p T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.g_bt_fw |
| 23 | $S_4\times C_2$ | \( 1 - T + 60 T^{2} - 49 T^{3} + 60 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ab_ci_abx |
| 29 | $S_4\times C_2$ | \( 1 - 13 T + 96 T^{2} - 505 T^{3} + 96 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.an_ds_atl |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 33 T^{2} - 28 T^{3} + 33 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ai_bh_abc |
| 37 | $S_4\times C_2$ | \( 1 - 11 T + 114 T^{2} - 643 T^{3} + 114 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.al_ek_ayt |
| 41 | $S_4\times C_2$ | \( 1 - 11 T + 152 T^{2} - 915 T^{3} + 152 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.al_fw_abjf |
| 43 | $S_4\times C_2$ | \( 1 + 13 T + 136 T^{2} + 935 T^{3} + 136 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.n_fg_bjz |
| 47 | $S_4\times C_2$ | \( 1 - T + 120 T^{2} - 97 T^{3} + 120 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ab_eq_adt |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 135 T^{2} - 704 T^{3} + 135 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ai_ff_abbc |
| 59 | $S_4\times C_2$ | \( 1 - 7 T + 68 T^{2} - 219 T^{3} + 68 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ah_cq_ail |
| 61 | $S_4\times C_2$ | \( 1 + 12 T + 123 T^{2} + 1340 T^{3} + 123 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.m_et_bzo |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 141 T^{2} + 604 T^{3} + 141 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.i_fl_xg |
| 71 | $S_4\times C_2$ | \( 1 - 20 T + 301 T^{2} - 2792 T^{3} + 301 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.au_lp_aedk |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 179 T^{2} + 568 T^{3} + 179 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.e_gx_vw |
| 79 | $S_4\times C_2$ | \( 1 + 27 T + 444 T^{2} + 4753 T^{3} + 444 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.bb_rc_hav |
| 83 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 2092 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.aq_jp_adcm |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) | 3.89.as_ol_afbo |
| 97 | $S_4\times C_2$ | \( 1 + 15 T + 258 T^{2} + 2251 T^{3} + 258 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.p_jy_dip |
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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
−8.200130138801226043797649761411, −7.60078424186168540975840557615, −7.44182968715486463624783571270, −7.30990854222680139761277628737, −6.71415192581902656936847399690, −6.52471724489796139912915816449, −6.44840909623073576276825233338, −6.11818522472653199760600187109, −5.86099993066881870296012963300, −5.79993577515767739659360182718, −5.35400836279207393231086022538, −4.97360845832762923880496027193, −4.65484656129505529893729940562, −4.49409784451730072376777458541, −4.28586932171741654735201054122, −3.80739852861789725542255588449, −3.37906022137469022199065834187, −3.10124369080828968200914530076, −2.90487428859727306149556357052, −2.49875142805317180810603258817, −2.15642629217040240916928780922, −2.07201929680190173092559968131, −1.21888084901437572043514111215, −0.911097251257613769412174644799, −0.41837354985317816800260393465,
0.41837354985317816800260393465, 0.911097251257613769412174644799, 1.21888084901437572043514111215, 2.07201929680190173092559968131, 2.15642629217040240916928780922, 2.49875142805317180810603258817, 2.90487428859727306149556357052, 3.10124369080828968200914530076, 3.37906022137469022199065834187, 3.80739852861789725542255588449, 4.28586932171741654735201054122, 4.49409784451730072376777458541, 4.65484656129505529893729940562, 4.97360845832762923880496027193, 5.35400836279207393231086022538, 5.79993577515767739659360182718, 5.86099993066881870296012963300, 6.11818522472653199760600187109, 6.44840909623073576276825233338, 6.52471724489796139912915816449, 6.71415192581902656936847399690, 7.30990854222680139761277628737, 7.44182968715486463624783571270, 7.60078424186168540975840557615, 8.200130138801226043797649761411
