| L(s) = 1 | + 2-s − 2·3-s − 2·4-s − 3·5-s − 2·6-s − 2·8-s − 9-s − 3·10-s + 2·11-s + 4·12-s + 2·13-s + 6·15-s + 16-s + 4·17-s − 18-s + 10·19-s + 6·20-s + 2·22-s + 16·23-s + 4·24-s + 6·25-s + 2·26-s + 4·27-s − 3·29-s + 6·30-s + 14·31-s − 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 4-s − 1.34·5-s − 0.816·6-s − 0.707·8-s − 1/3·9-s − 0.948·10-s + 0.603·11-s + 1.15·12-s + 0.554·13-s + 1.54·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 2.29·19-s + 1.34·20-s + 0.426·22-s + 3.33·23-s + 0.816·24-s + 6/5·25-s + 0.392·26-s + 0.769·27-s − 0.557·29-s + 1.09·30-s + 2.51·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 7^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.685315033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.685315033\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 7 | | \( 1 \) | |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - T + 3 T^{2} - 3 T^{3} + 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ab_d_ad |
| 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_f_i |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 25 T^{2} - 48 T^{3} + 25 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_z_abw |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_bb_abs |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + 11 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_l_acq |
| 19 | $S_4\times C_2$ | \( 1 - 10 T + 85 T^{2} - 400 T^{3} + 85 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ak_dh_apk |
| 23 | $S_4\times C_2$ | \( 1 - 16 T + 145 T^{2} - 36 p T^{3} + 145 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.aq_fp_abfw |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 153 T^{2} - 944 T^{3} + 153 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ao_fx_abki |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 87 T^{2} + 500 T^{3} + 87 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.i_dj_tg |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 39 T^{2} - 396 T^{3} + 39 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_bn_apg |
| 43 | $S_4\times C_2$ | \( 1 - 2 T - 3 T^{2} - 176 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_ad_agu |
| 47 | $S_4\times C_2$ | \( 1 + 14 T + 201 T^{2} + 1392 T^{3} + 201 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.o_ht_cbo |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 155 T^{2} - 628 T^{3} + 155 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ag_fz_aye |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.i_ej_bnk |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 75 T^{2} - 516 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ag_cx_atw |
| 67 | $S_4\times C_2$ | \( 1 - 28 T + 453 T^{2} - 4468 T^{3} + 453 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.abc_rl_agpw |
| 71 | $S_4\times C_2$ | \( 1 - 28 T + 389 T^{2} - 3704 T^{3} + 389 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.abc_oz_afmm |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 119 T^{2} - 636 T^{3} + 119 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aq_ep_aym |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 149 T^{2} + 488 T^{3} + 149 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.g_ft_su |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 1844 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.m_jp_csy |
| 89 | $S_4\times C_2$ | \( 1 - 10 T + 279 T^{2} - 1740 T^{3} + 279 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ak_kt_acoy |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 223 T^{2} + 1628 T^{3} + 223 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.i_ip_ckq |
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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
−7.07347124709586413116758430667, −6.75636173695311193559496704987, −6.49594097689542842098800646270, −6.41206504746946424421650534706, −6.01325154075514951564649969806, −5.54525432744848025520565310558, −5.43688721272347723904718830082, −5.24922504875110803954831920920, −5.16951855885689172223373278470, −4.94709771753851750178196703070, −4.68946100721651651806758754269, −4.52948114101990415023136264367, −4.06462994948761441162295704627, −3.72203777906503592039903505028, −3.59810000857623001231369606742, −3.57430356662902259225696961675, −3.04374273963547748302740249037, −3.04212942244126921001894949381, −2.53867280187488933798538628050, −2.35173584175881314080772769128, −1.44393377522837089334996124045, −1.21693149359206007935981309955, −1.04597890059242463270513278043, −0.60164466764709793396470255615, −0.43930095579971578481891079714,
0.43930095579971578481891079714, 0.60164466764709793396470255615, 1.04597890059242463270513278043, 1.21693149359206007935981309955, 1.44393377522837089334996124045, 2.35173584175881314080772769128, 2.53867280187488933798538628050, 3.04212942244126921001894949381, 3.04374273963547748302740249037, 3.57430356662902259225696961675, 3.59810000857623001231369606742, 3.72203777906503592039903505028, 4.06462994948761441162295704627, 4.52948114101990415023136264367, 4.68946100721651651806758754269, 4.94709771753851750178196703070, 5.16951855885689172223373278470, 5.24922504875110803954831920920, 5.43688721272347723904718830082, 5.54525432744848025520565310558, 6.01325154075514951564649969806, 6.41206504746946424421650534706, 6.49594097689542842098800646270, 6.75636173695311193559496704987, 7.07347124709586413116758430667
