| 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
| \(L(1)\) |
\(\approx\) |
\(3.086590531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.086590531\) |
| \(L(\frac{3}{2})\) |
|
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.c_e_g |
| 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.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.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.h_bs_gx |
| 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.j_do_qp |
| 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.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.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.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.au_kb_adci |
| 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.az_lc_adkn |
| 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.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.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.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.n_hk_bxn |
| 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.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
−7.33050542589917312070267887838, −6.82682696850298276513664953130, −6.78385775431751086821191809665, −6.24772604838185152067566077583, −6.04199944283146971207984025718, −5.78524899841209906941447868612, −5.50449785074213186716315092871, −5.41169259019449137488827309704, −5.09799785104529587554297385863, −5.03956304990157021347591470712, −4.49651785315267412896143355891, −4.36300199068358067643432436278, −4.24031518038589890799523540922, −3.80933434500313418211593239467, −3.77265112442117644263332286865, −3.37926872636505339908936220495, −2.85678209184448213077647367074, −2.38827376021713431231899321575, −2.28735092345474814453979899205, −2.07173092937544760936207927847, −1.77606513184899388648728545548, −1.38309922256865639846575437420, −0.885308570468560949032924052228, −0.61663328547200058508501289006, −0.60785939300198598557888226678,
0.60785939300198598557888226678, 0.61663328547200058508501289006, 0.885308570468560949032924052228, 1.38309922256865639846575437420, 1.77606513184899388648728545548, 2.07173092937544760936207927847, 2.28735092345474814453979899205, 2.38827376021713431231899321575, 2.85678209184448213077647367074, 3.37926872636505339908936220495, 3.77265112442117644263332286865, 3.80933434500313418211593239467, 4.24031518038589890799523540922, 4.36300199068358067643432436278, 4.49651785315267412896143355891, 5.03956304990157021347591470712, 5.09799785104529587554297385863, 5.41169259019449137488827309704, 5.50449785074213186716315092871, 5.78524899841209906941447868612, 6.04199944283146971207984025718, 6.24772604838185152067566077583, 6.78385775431751086821191809665, 6.82682696850298276513664953130, 7.33050542589917312070267887838
