| 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
| \(L(1)\) |
\(\approx\) |
\(0.6050837854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6050837854\) |
| \(L(\frac{3}{2})\) |
|
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.d_j_s |
| 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.ag_bk_aew |
| 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.d_bb_ek |
| 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.g_bt_gy |
| 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.d_bz_ic |
| 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.am_et_abjc |
| 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_ay |
| 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.m_gy_bws |
| 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.s_hh_cim |
| 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_aee |
| 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.am_jp_acvw |
| 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.p_md_dww |
| 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
−6.69504280818416313416537103272, −6.47604224495104633634657973574, −6.41330717923534364784981232476, −6.20888937799330238092424135034, −5.97510411393991675561677860594, −5.88984843318298918819345837827, −5.63191888007304403314612087365, −5.03312061058291507440279247540, −4.82290937381494165643438926728, −4.75878031235352901887486064311, −4.25830087079559740960235065821, −4.07218923075284181224995748484, −4.07155904715847037903652248967, −3.61524735732678535382807485345, −3.58436722854525025157214741942, −3.46040442092473619875381029380, −2.83926202285391421824777786978, −2.74232888262300562423694208891, −2.53406721479455700976467759211, −1.75766132116686449377106924302, −1.68743020828387458095695945510, −1.64486734006021153895460423731, −0.893110923689703656244495893820, −0.64337485459616299522836317372, −0.15863751109082976838183983655,
0.15863751109082976838183983655, 0.64337485459616299522836317372, 0.893110923689703656244495893820, 1.64486734006021153895460423731, 1.68743020828387458095695945510, 1.75766132116686449377106924302, 2.53406721479455700976467759211, 2.74232888262300562423694208891, 2.83926202285391421824777786978, 3.46040442092473619875381029380, 3.58436722854525025157214741942, 3.61524735732678535382807485345, 4.07155904715847037903652248967, 4.07218923075284181224995748484, 4.25830087079559740960235065821, 4.75878031235352901887486064311, 4.82290937381494165643438926728, 5.03312061058291507440279247540, 5.63191888007304403314612087365, 5.88984843318298918819345837827, 5.97510411393991675561677860594, 6.20888937799330238092424135034, 6.41330717923534364784981232476, 6.47604224495104633634657973574, 6.69504280818416313416537103272
