| L(s) = 1 | + 2·5-s + 2·7-s − 10·11-s − 2·13-s + 6·17-s + 12·19-s − 8·23-s − 3·25-s + 6·29-s − 4·31-s + 4·35-s + 2·37-s − 3·41-s + 8·43-s + 8·47-s − 3·49-s + 10·53-s − 20·55-s + 12·59-s + 10·61-s − 4·65-s + 6·67-s − 8·71-s + 10·73-s − 20·77-s − 8·79-s + 12·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 3.01·11-s − 0.554·13-s + 1.45·17-s + 2.75·19-s − 1.66·23-s − 3/5·25-s + 1.11·29-s − 0.718·31-s + 0.676·35-s + 0.328·37-s − 0.468·41-s + 1.21·43-s + 1.16·47-s − 3/7·49-s + 1.37·53-s − 2.69·55-s + 1.56·59-s + 1.28·61-s − 0.496·65-s + 0.733·67-s − 0.949·71-s + 1.17·73-s − 2.27·77-s − 0.900·79-s + 1.30·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.733832324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.733832324\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 41 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 T + 7 T^{2} - 24 T^{3} + 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ac_h_ay |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 38 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ac_h_abm |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 51 T^{2} + 186 T^{3} + 51 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.k_bz_he |
| 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.c_bb_bs |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.17.ag_cl_aie |
| 19 | $S_4\times C_2$ | \( 1 - 12 T + 71 T^{2} - 322 T^{3} + 71 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.am_ct_amk |
| 23 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 336 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.i_cr_my |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 83 T^{2} - 340 T^{3} + 83 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ag_df_anc |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 77 T^{2} + 216 T^{3} + 77 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.e_cz_ii |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 103 T^{2} - 152 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ac_dz_afw |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 97 T^{2} - 416 T^{3} + 97 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ai_dt_aqa |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 87 T^{2} - 546 T^{3} + 87 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ai_dj_ava |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 139 T^{2} - 1068 T^{3} + 139 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ak_fj_abpc |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 65 T^{2} - 232 T^{3} + 65 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.am_cn_aiy |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 131 T^{2} - 684 T^{3} + 131 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ak_fb_abai |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 179 T^{2} - 806 T^{3} + 179 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_gx_abfa |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 67 T^{2} + 546 T^{3} + 67 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.i_cp_va |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 155 T^{2} - 1368 T^{3} + 155 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ak_fz_acaq |
| 79 | $S_4\times C_2$ | \( 1 + 8 T + 71 T^{2} - 186 T^{3} + 71 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.i_ct_ahe |
| 83 | $S_4\times C_2$ | \( 1 - 7 T^{2} + 1024 T^{3} - 7 p T^{4} + p^{3} T^{6} \) | 3.83.a_ah_bnk |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 263 T^{2} + 2332 T^{3} + 263 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.s_kd_dls |
| 97 | $S_4\times C_2$ | \( 1 + 26 T + 463 T^{2} + 5228 T^{3} + 463 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ba_rv_htc |
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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
−7.39696964125345992621447901180, −7.04196568273508929954108304326, −6.78293244030638119652238910430, −6.47562229567023737956606909249, −5.82248191438174256964727628596, −5.77612412244206450765019983498, −5.64434675541846989040435697506, −5.52074061967489815181538068436, −5.33693452324972021499600944828, −5.24857213905469797398132675629, −4.69505212401489898207320541850, −4.47028843224931822540014036538, −4.44320221846754695003602927744, −3.77906264062451634700517982800, −3.62672460535101435368905325402, −3.39380326118926357023621737005, −2.81581736344766912505464876810, −2.71600872113457431494588874897, −2.67801980857831121995938786178, −2.15356751333882862612894983061, −1.92277514963795832165439104845, −1.66958136808711861909274259973, −0.927418284673985690289314951090, −0.922119159130548415737796342879, −0.34470930054876206659885108482,
0.34470930054876206659885108482, 0.922119159130548415737796342879, 0.927418284673985690289314951090, 1.66958136808711861909274259973, 1.92277514963795832165439104845, 2.15356751333882862612894983061, 2.67801980857831121995938786178, 2.71600872113457431494588874897, 2.81581736344766912505464876810, 3.39380326118926357023621737005, 3.62672460535101435368905325402, 3.77906264062451634700517982800, 4.44320221846754695003602927744, 4.47028843224931822540014036538, 4.69505212401489898207320541850, 5.24857213905469797398132675629, 5.33693452324972021499600944828, 5.52074061967489815181538068436, 5.64434675541846989040435697506, 5.77612412244206450765019983498, 5.82248191438174256964727628596, 6.47562229567023737956606909249, 6.78293244030638119652238910430, 7.04196568273508929954108304326, 7.39696964125345992621447901180
