| L(s) = 1 | + 3·2-s + 6·4-s − 3·5-s − 4·7-s + 10·8-s − 9·10-s − 2·11-s − 6·13-s − 12·14-s + 15·16-s + 4·17-s + 6·19-s − 18·20-s − 6·22-s + 6·25-s − 18·26-s − 24·28-s + 21·32-s + 12·34-s + 12·35-s − 8·37-s + 18·38-s − 30·40-s + 2·41-s − 4·43-s − 12·44-s − 2·47-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s − 1.34·5-s − 1.51·7-s + 3.53·8-s − 2.84·10-s − 0.603·11-s − 1.66·13-s − 3.20·14-s + 15/4·16-s + 0.970·17-s + 1.37·19-s − 4.02·20-s − 1.27·22-s + 6/5·25-s − 3.53·26-s − 4.53·28-s + 3.71·32-s + 2.05·34-s + 2.02·35-s − 1.31·37-s + 2.91·38-s − 4.74·40-s + 0.312·41-s − 0.609·43-s − 1.80·44-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 89^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 89^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 89 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 4 T + 23 T^{2} + 54 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_x_cc |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 29 T^{2} + 40 T^{3} + 29 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_bd_bo |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 47 T^{2} + 154 T^{3} + 47 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bv_fy |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 116 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bv_aem |
| 19 | $S_4\times C_2$ | \( 1 - 6 T + 41 T^{2} - 232 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ag_bp_aiy |
| 23 | $S_4\times C_2$ | \( 1 + 35 T^{2} + 62 T^{3} + 35 p T^{4} + p^{3} T^{6} \) | 3.23.a_bj_ck |
| 29 | $S_4\times C_2$ | \( 1 + 57 T^{2} + 2 T^{3} + 57 p T^{4} + p^{3} T^{6} \) | 3.29.a_cf_c |
| 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_cc |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 119 T^{2} + 594 T^{3} + 119 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.i_ep_ww |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 103 T^{2} - 156 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_dz_aga |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 97 T^{2} + 376 T^{3} + 97 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.e_dt_om |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 49 T^{2} - 12 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.c_bx_am |
| 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 | $S_4\times C_2$ | \( 1 + 10 T + 121 T^{2} + 624 T^{3} + 121 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.k_er_ya |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 189 T^{2} + 974 T^{3} + 189 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.i_hh_blm |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 145 T^{2} + 1052 T^{3} + 145 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.i_fp_bom |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 209 T^{2} + 1412 T^{3} + 209 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.k_ib_cci |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 319 T^{2} + 3068 T^{3} + 319 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.u_mh_eoa |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + T^{2} - 1228 T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.c_b_abvg |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 161 T^{2} + 680 T^{3} + 161 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.e_gf_bae |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 255 T^{2} + 2404 T^{3} + 255 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.w_jv_dom |
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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
−7.29297185338554934242551279480, −6.85023559007036045611912035013, −6.83690138969109287009325233819, −6.76951057899495688417529312504, −6.08291111116679509426267254761, −6.02931534295374192804334260546, −5.94057856482072353309133394904, −5.52216362677515401532728535334, −5.27551020156419619625364047014, −5.21802084805157564446095701064, −4.79398541429704373479808190850, −4.62103059479347345974953276981, −4.60169632464447481378627793221, −4.11668176731580012437287761504, −3.76527725520111037485235645808, −3.72881583889913935685371996922, −3.33838442502440855869423990785, −3.24763033159372594992107725852, −3.00912040627669135440897490493, −2.53108275490134355661082555868, −2.51221059443783216229135566018, −2.50638755524264294913397015683, −1.39755203886363460461665758566, −1.35155402899621973386472681764, −1.32069728349064582993588883939, 0, 0, 0,
1.32069728349064582993588883939, 1.35155402899621973386472681764, 1.39755203886363460461665758566, 2.50638755524264294913397015683, 2.51221059443783216229135566018, 2.53108275490134355661082555868, 3.00912040627669135440897490493, 3.24763033159372594992107725852, 3.33838442502440855869423990785, 3.72881583889913935685371996922, 3.76527725520111037485235645808, 4.11668176731580012437287761504, 4.60169632464447481378627793221, 4.62103059479347345974953276981, 4.79398541429704373479808190850, 5.21802084805157564446095701064, 5.27551020156419619625364047014, 5.52216362677515401532728535334, 5.94057856482072353309133394904, 6.02931534295374192804334260546, 6.08291111116679509426267254761, 6.76951057899495688417529312504, 6.83690138969109287009325233819, 6.85023559007036045611912035013, 7.29297185338554934242551279480
