| L(s) = 1 | − 9-s + 2·11-s − 2·13-s − 3·17-s − 8·23-s + 4·27-s − 2·29-s + 10·31-s + 6·37-s + 18·41-s + 10·43-s + 2·47-s − 13·49-s − 14·53-s − 16·59-s − 6·61-s + 26·67-s + 14·71-s + 10·73-s − 14·79-s − 2·81-s + 18·83-s + 26·89-s + 2·97-s − 2·99-s + 22·101-s − 2·103-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.727·17-s − 1.66·23-s + 0.769·27-s − 0.371·29-s + 1.79·31-s + 0.986·37-s + 2.81·41-s + 1.52·43-s + 0.291·47-s − 1.85·49-s − 1.92·53-s − 2.08·59-s − 0.768·61-s + 3.17·67-s + 1.66·71-s + 1.17·73-s − 1.57·79-s − 2/9·81-s + 1.97·83-s + 2.75·89-s + 0.203·97-s − 0.201·99-s + 2.18·101-s − 0.197·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.885796227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.885796227\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + T^{2} - 4 T^{3} + p T^{4} + p^{3} T^{6} \) | 3.3.a_b_ae |
| 7 | $S_4\times C_2$ | \( 1 + 13 T^{2} - 4 T^{3} + 13 p T^{4} + p^{3} T^{6} \) | 3.7.a_n_ae |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 17 T^{2} - 48 T^{3} + 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_r_abw |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 11 T^{2} - 20 T^{3} + 11 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.c_l_au |
| 19 | $S_4\times C_2$ | \( 1 + 25 T^{2} - 32 T^{3} + 25 p T^{4} + p^{3} T^{6} \) | 3.19.a_z_abg |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 21 T^{2} - 20 T^{3} + 21 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.i_v_au |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} - 52 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.c_t_aca |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 85 T^{2} - 576 T^{3} + 85 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ak_dh_awe |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 91 T^{2} - 420 T^{3} + 91 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_dn_aqe |
| 41 | $S_4\times C_2$ | \( 1 - 18 T + 199 T^{2} - 1468 T^{3} + 199 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.as_hr_acem |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 133 T^{2} - 852 T^{3} + 133 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ak_fd_abgu |
| 47 | $S_4\times C_2$ | \( 1 - 2 T + 121 T^{2} - 196 T^{3} + 121 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ac_er_aho |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 139 T^{2} + 1012 T^{3} + 139 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.o_fj_bmy |
| 59 | $S_4\times C_2$ | \( 1 + 16 T + 193 T^{2} + 1792 T^{3} + 193 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.q_hl_cqy |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 67 T^{2} + 740 T^{3} + 67 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.g_cp_bcm |
| 67 | $S_4\times C_2$ | \( 1 - 26 T + 405 T^{2} - 3972 T^{3} + 405 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.aba_pp_afwu |
| 71 | $S_4\times C_2$ | \( 1 - 14 T + 261 T^{2} - 2024 T^{3} + 261 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ao_kb_aczw |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 87 T^{2} - 492 T^{3} + 87 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ak_dj_asy |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 285 T^{2} + 2248 T^{3} + 285 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.o_kz_dim |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 205 T^{2} - 1604 T^{3} + 205 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.as_hx_acjs |
| 89 | $S_4\times C_2$ | \( 1 - 26 T + 391 T^{2} - 4124 T^{3} + 391 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.aba_pb_agcq |
| 97 | $S_4\times C_2$ | \( 1 - 2 T + 15 T^{2} - 1180 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ac_p_abtk |
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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
−8.323101122031535135992689778338, −7.87964397616345244459986508393, −7.69907296639068968580355156261, −7.65820961813170732702220081184, −7.47057285186789067553968725577, −6.76420787668635089358630389734, −6.57718259287202384773786020078, −6.28445398538175233499982441649, −6.22806516606467278883574168980, −6.05975568899445656721234724045, −5.58682961726094468498547187450, −5.19474885060528156668963521521, −4.89846217389089886790068370043, −4.46862928053709123760752947203, −4.39444127335342270433618548830, −4.29691507750850660810932108612, −3.60042695521575062905139585187, −3.32198403138992215161017740376, −3.19332314569623243538034289368, −2.37221463321489928342622351202, −2.36588436982769897425463435094, −2.15306909733460021715948235728, −1.45457895523075657036915804987, −0.826616203711609529673443946636, −0.53961782848774431033963339417,
0.53961782848774431033963339417, 0.826616203711609529673443946636, 1.45457895523075657036915804987, 2.15306909733460021715948235728, 2.36588436982769897425463435094, 2.37221463321489928342622351202, 3.19332314569623243538034289368, 3.32198403138992215161017740376, 3.60042695521575062905139585187, 4.29691507750850660810932108612, 4.39444127335342270433618548830, 4.46862928053709123760752947203, 4.89846217389089886790068370043, 5.19474885060528156668963521521, 5.58682961726094468498547187450, 6.05975568899445656721234724045, 6.22806516606467278883574168980, 6.28445398538175233499982441649, 6.57718259287202384773786020078, 6.76420787668635089358630389734, 7.47057285186789067553968725577, 7.65820961813170732702220081184, 7.69907296639068968580355156261, 7.87964397616345244459986508393, 8.323101122031535135992689778338
