| L(s) = 1 | − 2·3-s + 2·7-s − 3·9-s − 3·13-s + 6·17-s + 10·19-s − 4·21-s + 12·23-s + 10·27-s − 14·29-s + 8·31-s + 4·37-s + 6·39-s − 4·41-s + 4·43-s + 10·47-s − 9·49-s − 12·51-s + 16·53-s − 20·57-s − 2·59-s − 6·61-s − 6·63-s − 14·67-s − 24·69-s + 10·71-s + 12·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s − 9-s − 0.832·13-s + 1.45·17-s + 2.29·19-s − 0.872·21-s + 2.50·23-s + 1.92·27-s − 2.59·29-s + 1.43·31-s + 0.657·37-s + 0.960·39-s − 0.624·41-s + 0.609·43-s + 1.45·47-s − 9/7·49-s − 1.68·51-s + 2.19·53-s − 2.64·57-s − 0.260·59-s − 0.768·61-s − 0.755·63-s − 1.71·67-s − 2.88·69-s + 1.18·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.434912084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.434912084\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 10 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_h_k |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 32 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_n_abg |
| 11 | $S_4\times C_2$ | \( 1 + 29 T^{2} + 2 T^{3} + 29 p T^{4} + p^{3} T^{6} \) | 3.11.a_bd_c |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 164 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ag_bv_agi |
| 19 | $S_4\times C_2$ | \( 1 - 10 T + 77 T^{2} - 354 T^{3} + 77 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ak_cz_anq |
| 23 | $S_4\times C_2$ | \( 1 - 12 T + 83 T^{2} - 418 T^{3} + 83 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.am_df_aqc |
| 29 | $S_4\times C_2$ | \( 1 + 14 T + 147 T^{2} + 888 T^{3} + 147 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.o_fr_bie |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 77 T^{2} - 366 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ai_cz_aoc |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 83 T^{2} - 180 T^{3} + 83 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ae_df_agy |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 107 T^{2} + 296 T^{3} + 107 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.e_ed_lk |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 131 T^{2} - 342 T^{3} + 131 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ae_fb_ane |
| 47 | $S_4\times C_2$ | \( 1 - 10 T - 11 T^{2} + 504 T^{3} - 11 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ak_al_tk |
| 53 | $S_4\times C_2$ | \( 1 - 16 T + 183 T^{2} - 1680 T^{3} + 183 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aq_hb_acmq |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 113 T^{2} + 58 T^{3} + 113 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.c_ej_cg |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 167 T^{2} + 632 T^{3} + 167 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.g_gl_yi |
| 67 | $S_4\times C_2$ | \( 1 + 14 T + 257 T^{2} + 1944 T^{3} + 257 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.o_jx_cwu |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 133 T^{2} - 618 T^{3} + 133 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ak_fd_axu |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 167 T^{2} - 1148 T^{3} + 167 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_gl_abse |
| 79 | $S_4\times C_2$ | \( 1 - 28 T + 485 T^{2} - 5112 T^{3} + 485 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.abc_sr_ahoq |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 317 T^{2} - 3040 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.as_mf_aemy |
| 89 | $S_4\times C_2$ | \( 1 + 22 T + 343 T^{2} + 3732 T^{3} + 343 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.w_nf_fno |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 159 T^{2} + 452 T^{3} + 159 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.g_gd_rk |
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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
−7.32839906554495809082825281019, −6.86516467385663533639222847167, −6.80183591368915542718410821711, −6.75271637985841824483467459794, −6.02810896406902739879529535538, −5.95350122387772638602113389472, −5.78916894799127647272114439277, −5.39768431090918474990250049064, −5.32246567246577402423819907939, −5.26534968324128159954535579782, −4.83588814311359444634005715554, −4.82264723819836516129107104195, −4.37417923762330061582858184723, −3.98731280417522509923350220114, −3.62681024098139088141259001446, −3.42229165405153042325718890447, −3.05367434443562373557642040282, −2.81236942466662883431581157291, −2.81175889068195628792316238166, −2.12434094279731884813173109215, −1.91612307443972861726717862459, −1.42716562774658445141371102929, −0.901107742012471617632578978878, −0.76301129477627103111875864416, −0.50256126840154819080521779987,
0.50256126840154819080521779987, 0.76301129477627103111875864416, 0.901107742012471617632578978878, 1.42716562774658445141371102929, 1.91612307443972861726717862459, 2.12434094279731884813173109215, 2.81175889068195628792316238166, 2.81236942466662883431581157291, 3.05367434443562373557642040282, 3.42229165405153042325718890447, 3.62681024098139088141259001446, 3.98731280417522509923350220114, 4.37417923762330061582858184723, 4.82264723819836516129107104195, 4.83588814311359444634005715554, 5.26534968324128159954535579782, 5.32246567246577402423819907939, 5.39768431090918474990250049064, 5.78916894799127647272114439277, 5.95350122387772638602113389472, 6.02810896406902739879529535538, 6.75271637985841824483467459794, 6.80183591368915542718410821711, 6.86516467385663533639222847167, 7.32839906554495809082825281019
