| L(s) = 1 | + 2-s − 5·7-s + 11-s − 3·13-s − 5·14-s − 16-s + 7·17-s + 6·19-s + 22-s + 4·23-s − 3·26-s + 13·29-s + 7·31-s + 32-s + 7·34-s − 2·37-s + 6·38-s + 4·46-s − 7·47-s + 49-s + 9·53-s + 13·58-s − 3·59-s − 5·61-s + 7·62-s − 2·64-s + 3·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.88·7-s + 0.301·11-s − 0.832·13-s − 1.33·14-s − 1/4·16-s + 1.69·17-s + 1.37·19-s + 0.213·22-s + 0.834·23-s − 0.588·26-s + 2.41·29-s + 1.25·31-s + 0.176·32-s + 1.20·34-s − 0.328·37-s + 0.973·38-s + 0.589·46-s − 1.02·47-s + 1/7·49-s + 1.23·53-s + 1.70·58-s − 0.390·59-s − 0.640·61-s + 0.889·62-s − 1/4·64-s + 0.366·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.809422677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.809422677\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} - T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ab_b_ab |
| 7 | $S_4\times C_2$ | \( 1 + 5 T + 24 T^{2} + 67 T^{3} + 24 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.f_y_cp |
| 11 | $S_4\times C_2$ | \( 1 - T + 2 p T^{2} - 13 T^{3} + 2 p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ab_w_an |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 46 T^{2} - 211 T^{3} + 46 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ah_bu_aid |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 45 T^{2} - 224 T^{3} + 45 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ag_bt_aiq |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 37 T^{2} - 220 T^{3} + 37 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ae_bl_aim |
| 29 | $S_4\times C_2$ | \( 1 - 13 T + 106 T^{2} - 625 T^{3} + 106 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.an_ec_ayb |
| 31 | $S_4\times C_2$ | \( 1 - 7 T + 2 p T^{2} - 211 T^{3} + 2 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ah_ck_aid |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 75 T^{2} + 40 T^{3} + 75 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.c_cx_bo |
| 41 | $S_4\times C_2$ | \( 1 + 99 T^{2} + 36 T^{3} + 99 p T^{4} + p^{3} T^{6} \) | 3.41.a_dv_bk |
| 43 | $S_4\times C_2$ | \( 1 + 45 T^{2} + 164 T^{3} + 45 p T^{4} + p^{3} T^{6} \) | 3.43.a_bt_gi |
| 47 | $S_4\times C_2$ | \( 1 + 7 T + 16 T^{2} - 251 T^{3} + 16 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.h_q_ajr |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 162 T^{2} - 873 T^{3} + 162 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_gg_abhp |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 144 T^{2} + 237 T^{3} + 144 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.d_fo_jd |
| 61 | $S_4\times C_2$ | \( 1 + 5 T + 146 T^{2} + 497 T^{3} + 146 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.f_fq_td |
| 67 | $S_4\times C_2$ | \( 1 - 3 T + 42 T^{2} + 461 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ad_bq_rt |
| 71 | $S_4\times C_2$ | \( 1 - 18 T + 297 T^{2} - 2592 T^{3} + 297 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.as_ll_advs |
| 73 | $S_4\times C_2$ | \( 1 + 26 T + 435 T^{2} + 4360 T^{3} + 435 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ba_qt_gls |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 161 T^{2} - 796 T^{3} + 161 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ae_gf_abeq |
| 83 | $S_4\times C_2$ | \( 1 - 13 T + 172 T^{2} - 1819 T^{3} + 172 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.an_gq_acrz |
| 89 | $S_4\times C_2$ | \( 1 - 16 T + 331 T^{2} - 2896 T^{3} + 331 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.aq_mt_aehk |
| 97 | $S_4\times C_2$ | \( 1 + 26 T + 471 T^{2} + 5260 T^{3} + 471 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ba_sd_hui |
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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
−7.73091876510015113006148371630, −7.41933940991363070085535825813, −7.25349223904032809135032762182, −6.84372787529703616982671575816, −6.72434811769549076571613372992, −6.40067272512068328163361287287, −6.32864633295789023760335951904, −6.04302566780370652298468117093, −5.53294913530974242607358992540, −5.52419346723371425681826371196, −5.01848633273602571417277793620, −4.93763764069146382206933841678, −4.63119784638903492962737171724, −4.37964040857403907881256357822, −4.02432519620956184730878767004, −3.59829262522090540047051965298, −3.24585343515219416223942288593, −3.10036168738566512468196353762, −3.07450493117493379169868517393, −2.65334584583275512442396627370, −2.30713738610221585019262852588, −1.64108631587427349257750069112, −1.29118339889589022022408815377, −0.66627493134070933225244846132, −0.59929359344533769096080970042,
0.59929359344533769096080970042, 0.66627493134070933225244846132, 1.29118339889589022022408815377, 1.64108631587427349257750069112, 2.30713738610221585019262852588, 2.65334584583275512442396627370, 3.07450493117493379169868517393, 3.10036168738566512468196353762, 3.24585343515219416223942288593, 3.59829262522090540047051965298, 4.02432519620956184730878767004, 4.37964040857403907881256357822, 4.63119784638903492962737171724, 4.93763764069146382206933841678, 5.01848633273602571417277793620, 5.52419346723371425681826371196, 5.53294913530974242607358992540, 6.04302566780370652298468117093, 6.32864633295789023760335951904, 6.40067272512068328163361287287, 6.72434811769549076571613372992, 6.84372787529703616982671575816, 7.25349223904032809135032762182, 7.41933940991363070085535825813, 7.73091876510015113006148371630
