| L(s) = 1 | + 2-s + 3·3-s + 2·4-s + 3·6-s − 3·7-s − 2·8-s + 6·9-s + 4·11-s + 6·12-s + 2·13-s − 3·14-s − 3·16-s + 12·17-s + 6·18-s − 3·19-s − 9·21-s + 4·22-s − 8·23-s − 6·24-s − 7·25-s + 2·26-s + 10·27-s − 6·28-s − 8·29-s − 8·31-s − 9·32-s + 12·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 4-s + 1.22·6-s − 1.13·7-s − 0.707·8-s + 2·9-s + 1.20·11-s + 1.73·12-s + 0.554·13-s − 0.801·14-s − 3/4·16-s + 2.91·17-s + 1.41·18-s − 0.688·19-s − 1.96·21-s + 0.852·22-s − 1.66·23-s − 1.22·24-s − 7/5·25-s + 0.392·26-s + 1.92·27-s − 1.13·28-s − 1.48·29-s − 1.43·31-s − 1.59·32-s + 2.08·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63521199 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63521199 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.821561813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.821561813\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - T - T^{2} + 5 T^{3} - p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ab_ab_f |
| 5 | $S_4\times C_2$ | \( 1 + 7 T^{2} - 4 T^{3} + 7 p T^{4} + p^{3} T^{6} \) | 3.5.a_h_ae |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 40 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ae_r_abo |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 60 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_t_aci |
| 17 | $S_4\times C_2$ | \( 1 - 12 T + 91 T^{2} - 436 T^{3} + 91 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.am_dn_aqu |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 352 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.i_cr_no |
| 29 | $S_4\times C_2$ | \( 1 + 8 T + 51 T^{2} + 172 T^{3} + 51 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.i_bz_gq |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 85 T^{2} + 384 T^{3} + 85 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.i_dh_ou |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} - 84 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.c_bb_adg |
| 41 | $S_4\times C_2$ | \( 1 - 22 T + 263 T^{2} - 2020 T^{3} + 263 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aw_kd_aczs |
| 43 | $S_4\times C_2$ | \( 1 - 16 T + 185 T^{2} - 1424 T^{3} + 185 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aq_hd_accu |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 85 T^{2} - 8 T^{3} + 85 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.c_dh_ai |
| 53 | $S_4\times C_2$ | \( 1 + 3 T^{2} - 412 T^{3} + 3 p T^{4} + p^{3} T^{6} \) | 3.53.a_d_apw |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 97 T^{2} - 712 T^{3} + 97 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.am_dt_abbk |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 163 T^{2} - 1236 T^{3} + 163 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ao_gh_abvo |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 153 T^{2} + 784 T^{3} + 153 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.i_fx_bee |
| 71 | $S_4\times C_2$ | \( 1 + 26 T + 369 T^{2} + 3520 T^{3} + 369 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ba_of_ffk |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 199 T^{2} - 788 T^{3} + 199 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ag_hr_abei |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 157 T^{2} - 1192 T^{3} + 157 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.am_gb_abtw |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 209 T^{2} - 248 T^{3} + 209 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ac_ib_ajo |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 71 T^{2} - 820 T^{3} + 71 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.c_ct_abfo |
| 97 | $S_4\times C_2$ | \( 1 - 22 T + 335 T^{2} - 3380 T^{3} + 335 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.aw_mx_afaa |
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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
−10.10098019668882453198405489037, −9.577275795515073125733009438514, −9.377628409531005987904558770660, −9.234744011646566970448844250378, −9.057157262933332291332071696904, −8.499316451876551641603028433308, −8.218743982575729610486258785953, −7.80351253833771637033469540537, −7.45045916428600013831258248695, −7.26177513710809638175320118686, −7.16969551988289074989399687874, −6.26386634874263879136623074131, −6.08452394566900666089326402371, −6.07651220312096780182276265831, −5.53369722505239456232946994510, −5.48811314175446666034803490128, −4.28894803860738802833264060397, −4.02050738556165835494517690274, −3.84103342289499534885613457887, −3.50661709752168825873656963691, −3.24014193388489653431117946010, −2.70116072859622686162844330461, −2.20679378543428829233892409743, −1.92556801346990797493138724393, −0.974093960119881868088782900473,
0.974093960119881868088782900473, 1.92556801346990797493138724393, 2.20679378543428829233892409743, 2.70116072859622686162844330461, 3.24014193388489653431117946010, 3.50661709752168825873656963691, 3.84103342289499534885613457887, 4.02050738556165835494517690274, 4.28894803860738802833264060397, 5.48811314175446666034803490128, 5.53369722505239456232946994510, 6.07651220312096780182276265831, 6.08452394566900666089326402371, 6.26386634874263879136623074131, 7.16969551988289074989399687874, 7.26177513710809638175320118686, 7.45045916428600013831258248695, 7.80351253833771637033469540537, 8.218743982575729610486258785953, 8.499316451876551641603028433308, 9.057157262933332291332071696904, 9.234744011646566970448844250378, 9.377628409531005987904558770660, 9.577275795515073125733009438514, 10.10098019668882453198405489037
