| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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.i_bl_em |
| 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.d_bg_cn |
| 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.e_x_ek |
| 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.j_dc_tp |
| 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.i_ep_ww |
| 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.c_cn_lm |
| 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.e_bx_ku |
| 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.c_dz_gg |
| 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.p_jc_czj |
| 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.am_dx_aua |
| 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.c_ct_wq |
| 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.ap_na_aecr |
| 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.h_kc_bwh |
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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
−7.21212712800466209942020080548, −7.06757445413840345543504870411, −6.77417623031062437807610965152, −6.73268634419593107283298299247, −6.36522849162830753863617789830, −6.10153987074547883429022969172, −6.06115027883150622638499407881, −5.43214016307418270573119207609, −5.34407686423941954023354951601, −5.26950768638567281063954418062, −5.00094147507387132943213657491, −4.89499783242131668669954485326, −4.70439761082637466824753834034, −4.12356493109902339675059513726, −3.76377227812448171605497472999, −3.76364131549396822913844997766, −3.32079356843485175649393577733, −3.20712237580641484667379486717, −3.06462596065945474727588387731, −2.82979188933678171018567613484, −2.81407668311107737005337219282, −1.94669149683860584898324320646, −1.77826372986396874959286586911, −1.15357686203623579087452051311, −1.03572161951298536661248239314, 0, 0, 0,
1.03572161951298536661248239314, 1.15357686203623579087452051311, 1.77826372986396874959286586911, 1.94669149683860584898324320646, 2.81407668311107737005337219282, 2.82979188933678171018567613484, 3.06462596065945474727588387731, 3.20712237580641484667379486717, 3.32079356843485175649393577733, 3.76364131549396822913844997766, 3.76377227812448171605497472999, 4.12356493109902339675059513726, 4.70439761082637466824753834034, 4.89499783242131668669954485326, 5.00094147507387132943213657491, 5.26950768638567281063954418062, 5.34407686423941954023354951601, 5.43214016307418270573119207609, 6.06115027883150622638499407881, 6.10153987074547883429022969172, 6.36522849162830753863617789830, 6.73268634419593107283298299247, 6.77417623031062437807610965152, 7.06757445413840345543504870411, 7.21212712800466209942020080548
