| L(s) = 1 | + 3·5-s − 3·7-s − 6·11-s + 3·13-s + 3·17-s + 6·23-s + 3·29-s − 6·31-s − 9·35-s + 15·37-s − 12·41-s − 12·43-s + 6·49-s + 12·53-s − 18·55-s + 18·59-s − 3·61-s + 9·65-s − 6·67-s + 9·73-s + 18·77-s − 6·79-s − 12·83-s + 9·85-s + 15·89-s − 9·91-s + 12·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s − 1.80·11-s + 0.832·13-s + 0.727·17-s + 1.25·23-s + 0.557·29-s − 1.07·31-s − 1.52·35-s + 2.46·37-s − 1.87·41-s − 1.82·43-s + 6/7·49-s + 1.64·53-s − 2.42·55-s + 2.34·59-s − 0.384·61-s + 1.11·65-s − 0.733·67-s + 1.05·73-s + 2.05·77-s − 0.675·79-s − 1.31·83-s + 0.976·85-s + 1.58·89-s − 0.943·91-s + 1.21·97-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
| \(L(1)\) |
\(\approx\) |
\(2.663064958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.663064958\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - 3 T + 9 T^{2} - 18 T^{3} + 9 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ad_j_as |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 126 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.g_bk_ew |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 34 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_d_bi |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 27 T^{2} - 114 T^{3} + 27 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ad_bb_aek |
| 19 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 56 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.19.a_j_ce |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 45 T^{2} - 180 T^{3} + 45 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ag_bt_agy |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 210 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_bz_aic |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.31.g_eb_oq |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{3} \) | 3.37.ap_he_abvn |
| 41 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 912 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.m_et_bjc |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 84 T^{2} + 380 T^{3} + 84 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.m_dg_oq |
| 47 | $S_4\times C_2$ | \( 1 + 117 T^{2} + 24 T^{3} + 117 p T^{4} + p^{3} T^{6} \) | 3.47.a_en_y |
| 53 | $S_4\times C_2$ | \( 1 - 12 T + 180 T^{2} - 1266 T^{3} + 180 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.am_gy_abws |
| 59 | $S_4\times C_2$ | \( 1 - 18 T + 189 T^{2} - 1572 T^{3} + 189 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.as_hh_acim |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 178 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.d_eb_gw |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 132 T^{2} + 542 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_fc_uw |
| 71 | $S_4\times C_2$ | \( 1 + 132 T^{2} + 108 T^{3} + 132 p T^{4} + p^{3} T^{6} \) | 3.71.a_fc_ee |
| 73 | $S_4\times C_2$ | \( 1 - 9 T + 207 T^{2} - 1298 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aj_hz_abxy |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 168 T^{2} + 686 T^{3} + 168 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.g_gm_bak |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 1920 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.m_jp_cvw |
| 89 | $S_4\times C_2$ | \( 1 - 15 T + 315 T^{2} - 2622 T^{3} + 315 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ap_md_adww |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 2144 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.am_lf_adem |
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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
−6.80040908828266490278950233223, −6.54199186418154863981745514726, −6.46494991836705844270774626487, −6.13055349884249246890422173344, −5.70914753648950583723172091884, −5.68607135561590157152592725256, −5.55695407163218800717503968616, −5.26692899643042423775048635602, −5.11378980200391631860259624654, −4.90492249644011997956352625975, −4.46398095100652495967404093868, −4.08566788142085652449226031233, −4.06948330256650261917577659782, −3.62111422133980327666611344687, −3.38900362857211006837166984012, −3.12996759578143054713093254344, −2.77258087111734362538728486548, −2.62649921443254413947878789244, −2.61817069014921301167823449866, −1.87215573890675755727360959957, −1.84636746054453646782138003880, −1.57740256636220867704906572342, −0.880252214247912605781906273632, −0.808684902307080792313719998769, −0.26424483479941972698299773125,
0.26424483479941972698299773125, 0.808684902307080792313719998769, 0.880252214247912605781906273632, 1.57740256636220867704906572342, 1.84636746054453646782138003880, 1.87215573890675755727360959957, 2.61817069014921301167823449866, 2.62649921443254413947878789244, 2.77258087111734362538728486548, 3.12996759578143054713093254344, 3.38900362857211006837166984012, 3.62111422133980327666611344687, 4.06948330256650261917577659782, 4.08566788142085652449226031233, 4.46398095100652495967404093868, 4.90492249644011997956352625975, 5.11378980200391631860259624654, 5.26692899643042423775048635602, 5.55695407163218800717503968616, 5.68607135561590157152592725256, 5.70914753648950583723172091884, 6.13055349884249246890422173344, 6.46494991836705844270774626487, 6.54199186418154863981745514726, 6.80040908828266490278950233223
