| L(s) = 1 | + 2·2-s + 8·7-s − 2·8-s + 3·11-s + 4·13-s + 16·14-s + 7·17-s + 9·19-s + 6·22-s + 9·23-s + 8·26-s + 9·29-s + 3·31-s + 14·34-s + 8·37-s + 18·38-s + 2·41-s + 4·43-s + 18·46-s − 20·47-s + 27·49-s − 25·53-s − 16·56-s + 18·58-s + 2·59-s + 2·61-s + 6·62-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3.02·7-s − 0.707·8-s + 0.904·11-s + 1.10·13-s + 4.27·14-s + 1.69·17-s + 2.06·19-s + 1.27·22-s + 1.87·23-s + 1.56·26-s + 1.67·29-s + 0.538·31-s + 2.40·34-s + 1.31·37-s + 2.91·38-s + 0.312·41-s + 0.609·43-s + 2.65·46-s − 2.91·47-s + 27/7·49-s − 3.43·53-s − 2.13·56-s + 2.36·58-s + 0.260·59-s + 0.256·61-s + 0.762·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(29.80154337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.80154337\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.2.ac_e_ag |
| 7 | $S_4\times C_2$ | \( 1 - 8 T + 37 T^{2} - 116 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ai_bl_aem |
| 11 | $S_4\times C_2$ | \( 1 - 3 T + 32 T^{2} - 65 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ad_bg_acn |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 23 T^{2} - 114 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ae_x_aek |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 44 T^{2} - 179 T^{3} + 44 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ah_bs_agx |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) | 3.19.aj_dg_aof |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 4 p T^{2} - 431 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.aj_do_aqp |
| 29 | $S_4\times C_2$ | \( 1 - 9 T + 80 T^{2} - 509 T^{3} + 80 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.aj_dc_atp |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 119 T^{2} - 594 T^{3} + 119 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_ep_aww |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 65 T^{2} - 298 T^{3} + 65 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_cn_alm |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 49 T^{2} - 280 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ae_bx_aku |
| 47 | $S_4\times C_2$ | \( 1 + 20 T + 261 T^{2} + 2088 T^{3} + 261 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.u_kb_dci |
| 53 | $S_4\times C_2$ | \( 1 + 25 T + 288 T^{2} + 2301 T^{3} + 288 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.z_lc_dkn |
| 59 | $S_4\times C_2$ | \( 1 - 2 T + 103 T^{2} - 162 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ac_dz_agg |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ac_fb_ahw |
| 67 | $S_4\times C_2$ | \( 1 - 15 T + 236 T^{2} - 2011 T^{3} + 236 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ap_jc_aczj |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 101 T^{2} + 520 T^{3} + 101 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.m_dx_ua |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} - 588 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ac_ct_awq |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1982 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.am_gj_acyg |
| 83 | $S_4\times C_2$ | \( 1 - 13 T + 192 T^{2} - 1287 T^{3} + 192 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.an_hk_abxn |
| 89 | $S_4\times C_2$ | \( 1 + 15 T + 338 T^{2} + 2773 T^{3} + 338 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.p_na_ecr |
| 97 | $S_4\times C_2$ | \( 1 - 7 T + 262 T^{2} - 1255 T^{3} + 262 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ah_kc_abwh |
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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
−6.83194404674129660988140610288, −6.70306402999765937284655205749, −6.46456739795887753623071748428, −6.41210497306705749058359593830, −5.91377007528447839558905885087, −5.56277098300067267464689802809, −5.53713000110371509065527774245, −5.13034799618978336557976235400, −5.04012858684929697246448628048, −4.90452923707909496770607163491, −4.59967453287364906382642584140, −4.53633858782277605786725323643, −4.37027647359501638272959322252, −3.73829937527720935332337308430, −3.69063758427273026670407214214, −3.49205603705608006757873349793, −3.02440323917632600493615033125, −2.96288527799279813563781917264, −2.66261952773628722489875742949, −2.03938819172001083747075530151, −1.59008523341475657856302003869, −1.46113299023059669979016579440, −1.22773401232409795920311725819, −0.970404856082068703393086216171, −0.69315674954118887785984302161,
0.69315674954118887785984302161, 0.970404856082068703393086216171, 1.22773401232409795920311725819, 1.46113299023059669979016579440, 1.59008523341475657856302003869, 2.03938819172001083747075530151, 2.66261952773628722489875742949, 2.96288527799279813563781917264, 3.02440323917632600493615033125, 3.49205603705608006757873349793, 3.69063758427273026670407214214, 3.73829937527720935332337308430, 4.37027647359501638272959322252, 4.53633858782277605786725323643, 4.59967453287364906382642584140, 4.90452923707909496770607163491, 5.04012858684929697246448628048, 5.13034799618978336557976235400, 5.53713000110371509065527774245, 5.56277098300067267464689802809, 5.91377007528447839558905885087, 6.41210497306705749058359593830, 6.46456739795887753623071748428, 6.70306402999765937284655205749, 6.83194404674129660988140610288
