| L(s) = 1 | − 3·3-s − 3·5-s − 2·7-s + 6·9-s + 12·11-s + 4·13-s + 9·15-s − 2·17-s + 6·21-s + 4·23-s + 6·25-s − 10·27-s − 4·29-s − 3·31-s − 36·33-s + 6·35-s + 8·37-s − 12·39-s − 6·41-s − 12·43-s − 18·45-s + 8·47-s − 49-s + 6·51-s − 14·53-s − 36·55-s + 6·59-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 0.755·7-s + 2·9-s + 3.61·11-s + 1.10·13-s + 2.32·15-s − 0.485·17-s + 1.30·21-s + 0.834·23-s + 6/5·25-s − 1.92·27-s − 0.742·29-s − 0.538·31-s − 6.26·33-s + 1.01·35-s + 1.31·37-s − 1.92·39-s − 0.937·41-s − 1.82·43-s − 2.68·45-s + 1.16·47-s − 1/7·49-s + 0.840·51-s − 1.92·53-s − 4.85·55-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.524234426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.524234426\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $D_{6}$ | \( 1 + 2 T + 5 T^{2} + 32 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_f_bg |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.11.am_dd_amq |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 27 T^{2} - 68 T^{3} + 27 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ae_bb_acq |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 31 T^{2} + 76 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.c_bf_cy |
| 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 | $S_4\times C_2$ | \( 1 - 4 T + 45 T^{2} - 200 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ae_bt_ahs |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 7 T^{2} + 20 T^{3} + 7 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.e_h_u |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 91 T^{2} - 12 p T^{3} + 91 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_dn_arc |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 12 T^{3} + 7 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.g_h_am |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 145 T^{2} + 1000 T^{3} + 145 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.m_fp_bmm |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 85 T^{2} - 320 T^{3} + 85 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ai_dh_ami |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 203 T^{2} + 1508 T^{3} + 203 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.o_hv_cga |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 181 T^{2} - 704 T^{3} + 181 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_gz_abbc |
| 61 | $S_4\times C_2$ | \( 1 + 10 T + 147 T^{2} + 1148 T^{3} + 147 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.k_fr_bse |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 145 T^{2} - 72 T^{3} + 145 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ac_fp_acu |
| 71 | $S_4\times C_2$ | \( 1 - 18 T + 313 T^{2} - 2728 T^{3} + 313 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.as_mb_aeay |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 311 T^{2} + 2884 T^{3} + 311 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.u_lz_egy |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 229 T^{2} - 1152 T^{3} + 229 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ai_iv_absi |
| 83 | $S_4\times C_2$ | \( 1 - 47 T^{2} - 528 T^{3} - 47 p T^{4} + p^{3} T^{6} \) | 3.83.a_abv_aui |
| 89 | $S_4\times C_2$ | \( 1 + 195 T^{2} + 108 T^{3} + 195 p T^{4} + p^{3} T^{6} \) | 3.89.a_hn_ee |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 63 T^{2} + 1500 T^{3} + 63 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.c_cl_cfs |
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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
−7.47360300432223215915002678200, −6.97764726237761041687414354226, −6.96259514243738963069258792249, −6.82738714243230785922084062195, −6.45303306214654406598695459435, −6.33009998171823856492027622091, −6.30158269420045165039754576734, −5.81583650230677779463392578346, −5.51599317071684454436331871965, −5.49941719735364674580747788190, −4.80640147779905441950500525545, −4.60014588150219937990208269205, −4.53075467183050083571109774689, −4.12820344070545622014612126270, −3.94883245998130379355947521445, −3.76986000468972610681989756404, −3.31270439743374466248981479618, −3.13088705305821151713935985365, −3.11842989425214552146938726617, −2.01635205396456739665054961796, −1.71281293109124348627785315021, −1.65943997627586665328910056825, −0.948987686877017089214317486225, −0.831369387683166627415918848051, −0.38112680126029452114395850467,
0.38112680126029452114395850467, 0.831369387683166627415918848051, 0.948987686877017089214317486225, 1.65943997627586665328910056825, 1.71281293109124348627785315021, 2.01635205396456739665054961796, 3.11842989425214552146938726617, 3.13088705305821151713935985365, 3.31270439743374466248981479618, 3.76986000468972610681989756404, 3.94883245998130379355947521445, 4.12820344070545622014612126270, 4.53075467183050083571109774689, 4.60014588150219937990208269205, 4.80640147779905441950500525545, 5.49941719735364674580747788190, 5.51599317071684454436331871965, 5.81583650230677779463392578346, 6.30158269420045165039754576734, 6.33009998171823856492027622091, 6.45303306214654406598695459435, 6.82738714243230785922084062195, 6.96259514243738963069258792249, 6.97764726237761041687414354226, 7.47360300432223215915002678200
