| L(s) = 1 | − 3·5-s + 2·7-s − 5·9-s − 4·11-s + 6·13-s + 10·17-s + 18·19-s − 2·23-s + 6·25-s + 2·27-s − 2·29-s − 6·35-s − 3·37-s − 6·41-s + 10·43-s + 15·45-s + 14·47-s + 3·49-s + 6·53-s + 12·55-s + 8·59-s + 10·61-s − 10·63-s − 18·65-s + 20·67-s − 32·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.755·7-s − 5/3·9-s − 1.20·11-s + 1.66·13-s + 2.42·17-s + 4.12·19-s − 0.417·23-s + 6/5·25-s + 0.384·27-s − 0.371·29-s − 1.01·35-s − 0.493·37-s − 0.937·41-s + 1.52·43-s + 2.23·45-s + 2.04·47-s + 3/7·49-s + 0.824·53-s + 1.61·55-s + 1.04·59-s + 1.28·61-s − 1.25·63-s − 2.23·65-s + 2.44·67-s − 3.79·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.256570535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256570535\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.3.a_f_ac |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + T^{2} + 22 T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_b_w |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 72 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.e_z_cu |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 23 T^{2} - 56 T^{3} + 23 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ag_x_ace |
| 17 | $S_4\times C_2$ | \( 1 - 10 T + 3 p T^{2} - 192 T^{3} + 3 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ak_bz_ahk |
| 19 | $S_4\times C_2$ | \( 1 - 18 T + 161 T^{2} - 878 T^{3} + 161 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.as_gf_abhu |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 57 T^{2} + 84 T^{3} + 57 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.c_cf_dg |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} - 68 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.c_bj_acq |
| 31 | $S_4\times C_2$ | \( 1 + 57 T^{2} - 54 T^{3} + 57 p T^{4} + p^{3} T^{6} \) | 3.31.a_cf_acc |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 119 T^{2} + 452 T^{3} + 119 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.g_ep_rk |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 109 T^{2} - 868 T^{3} + 109 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ak_ef_abhk |
| 47 | $S_4\times C_2$ | \( 1 - 14 T + 3 p T^{2} - 1242 T^{3} + 3 p^{2} T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ao_fl_abvu |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 59 T^{2} - 836 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_ch_abge |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 85 T^{2} - 786 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ai_dh_abeg |
| 61 | $S_4\times C_2$ | \( 1 - 10 T - 29 T^{2} + 836 T^{3} - 29 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ak_abd_bge |
| 67 | $S_4\times C_2$ | \( 1 - 20 T + 153 T^{2} - 830 T^{3} + 153 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.au_fx_abfy |
| 71 | $S_4\times C_2$ | \( 1 + 32 T + 517 T^{2} + 5280 T^{3} + 517 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.bg_tx_hvc |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 207 T^{2} - 284 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ac_hz_aky |
| 79 | $S_4\times C_2$ | \( 1 + 8 T + 197 T^{2} + 1002 T^{3} + 197 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.i_hp_bmo |
| 83 | $S_4\times C_2$ | \( 1 + 16 T + 313 T^{2} + 2730 T^{3} + 313 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.q_mb_eba |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 852 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_dj_bgu |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 383 T^{2} - 3628 T^{3} + 383 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.as_ot_afjo |
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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
−9.219766453059109621326389419320, −8.693686070883943946110029143886, −8.606031445157603895972500082029, −8.445982142185202657274368423521, −7.88732067957471548527583511297, −7.86494130682760323250515299072, −7.63915800517942326986221648021, −7.32946555558656681313014232762, −7.05114114867681419891688651742, −6.77499573455577936889579889750, −5.88067081928386728704812378843, −5.68333533225066731175895703934, −5.64537447012081222041193915883, −5.44767310540767139211209495284, −5.10884531878253907202826604715, −4.74106964991201324251356936378, −3.98614568204043363621333769692, −3.72272476149387066681604674080, −3.64623831715199675677521248447, −3.03950563740928009722648474787, −2.78548965009449560310630848329, −2.68234812895512438218964964944, −1.47808082823774441614281911443, −1.09579532768193590386799512452, −0.68024316918978915770171989201,
0.68024316918978915770171989201, 1.09579532768193590386799512452, 1.47808082823774441614281911443, 2.68234812895512438218964964944, 2.78548965009449560310630848329, 3.03950563740928009722648474787, 3.64623831715199675677521248447, 3.72272476149387066681604674080, 3.98614568204043363621333769692, 4.74106964991201324251356936378, 5.10884531878253907202826604715, 5.44767310540767139211209495284, 5.64537447012081222041193915883, 5.68333533225066731175895703934, 5.88067081928386728704812378843, 6.77499573455577936889579889750, 7.05114114867681419891688651742, 7.32946555558656681313014232762, 7.63915800517942326986221648021, 7.86494130682760323250515299072, 7.88732067957471548527583511297, 8.445982142185202657274368423521, 8.606031445157603895972500082029, 8.693686070883943946110029143886, 9.219766453059109621326389419320
