| L(s) = 1 | + 3·3-s − 3·5-s + 3·9-s + 6·11-s − 9·15-s + 9·17-s + 15·19-s − 6·23-s + 6·25-s + 2·27-s − 3·31-s + 18·33-s + 9·37-s + 3·41-s + 3·43-s − 9·45-s + 6·47-s − 9·49-s + 27·51-s − 21·53-s − 18·55-s + 45·57-s − 9·59-s − 12·61-s + 6·67-s − 18·69-s + 9·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.34·5-s + 9-s + 1.80·11-s − 2.32·15-s + 2.18·17-s + 3.44·19-s − 1.25·23-s + 6/5·25-s + 0.384·27-s − 0.538·31-s + 3.13·33-s + 1.47·37-s + 0.468·41-s + 0.457·43-s − 1.34·45-s + 0.875·47-s − 9/7·49-s + 3.78·51-s − 2.88·53-s − 2.42·55-s + 5.96·57-s − 1.17·59-s − 1.53·61-s + 0.733·67-s − 2.16·69-s + 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.505162761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.505162761\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - p T + 2 p T^{2} - 11 T^{3} + 2 p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.3.ad_g_al |
| 7 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 12 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.7.a_j_am |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 104 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ag_bh_aea |
| 13 | $S_4\times C_2$ | \( 1 + 27 T^{2} - 12 T^{3} + 27 p T^{4} + p^{3} T^{6} \) | 3.13.a_bb_am |
| 17 | $S_4\times C_2$ | \( 1 - 9 T + 66 T^{2} - 299 T^{3} + 66 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.aj_co_aln |
| 19 | $S_4\times C_2$ | \( 1 - 15 T + 120 T^{2} - 623 T^{3} + 120 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ap_eq_axz |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 57 T^{2} + 280 T^{3} + 57 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_cf_ku |
| 29 | $S_4\times C_2$ | \( 1 + 63 T^{2} - 44 T^{3} + 63 p T^{4} + p^{3} T^{6} \) | 3.29.a_cl_abs |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 114 T^{2} - 595 T^{3} + 114 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.aj_ek_awx |
| 41 | $S_4\times C_2$ | \( 1 - 3 T + 30 T^{2} - 279 T^{3} + 30 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ad_be_akt |
| 43 | $S_4\times C_2$ | \( 1 - 3 T + 151 T^{3} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ad_a_fv |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 540 T^{3} + 129 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ag_ez_auu |
| 53 | $S_4\times C_2$ | \( 1 + 21 T + 240 T^{2} + 1969 T^{3} + 240 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.v_jg_cxt |
| 59 | $S_4\times C_2$ | \( 1 + 9 T + 180 T^{2} + 1033 T^{3} + 180 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.j_gy_bnt |
| 61 | $S_4\times C_2$ | \( 1 + 12 T + 207 T^{2} + 1416 T^{3} + 207 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.m_hz_ccm |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 448 T^{3} + 129 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_ez_arg |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 48 T^{2} - 381 T^{3} + 48 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_bw_aor |
| 73 | $S_4\times C_2$ | \( 1 + 3 T + 48 T^{2} + 1131 T^{3} + 48 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.d_bw_brn |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 153 T^{2} - 884 T^{3} + 153 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.am_fx_abia |
| 83 | $S_4\times C_2$ | \( 1 + 21 T + 372 T^{2} + 3687 T^{3} + 372 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.v_oi_flv |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 147 T^{2} + 264 T^{3} + 147 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_fr_ke |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 351 T^{2} - 3436 T^{3} + 351 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.as_nn_afce |
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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
−9.551534676747281942827720215751, −9.283874765210793305804957032726, −8.804250593409703629129334147213, −8.656173720538163881522305901026, −8.093569712669832246212012715761, −7.925541553474182287446286121927, −7.75561733435882782922605833583, −7.61668218273475816045616360166, −7.33559322371250740489207306901, −6.94911732690747474385746589615, −6.43668514439369478537176311986, −6.07562995995735134295235742793, −5.93649503939031709028667927699, −5.27730578331007567162264642383, −5.07060581474082825910441812281, −4.67480273482346826378612563843, −3.99448247593365503690981698372, −3.86930665192893372684080345028, −3.67678324889039567497890175119, −3.00455886170072183236959753320, −2.98774655095644595384053136580, −2.86578005761319480366619754986, −1.64053283404703884584644514616, −1.41456568886331644210687001680, −0.844520332064857082068299234859,
0.844520332064857082068299234859, 1.41456568886331644210687001680, 1.64053283404703884584644514616, 2.86578005761319480366619754986, 2.98774655095644595384053136580, 3.00455886170072183236959753320, 3.67678324889039567497890175119, 3.86930665192893372684080345028, 3.99448247593365503690981698372, 4.67480273482346826378612563843, 5.07060581474082825910441812281, 5.27730578331007567162264642383, 5.93649503939031709028667927699, 6.07562995995735134295235742793, 6.43668514439369478537176311986, 6.94911732690747474385746589615, 7.33559322371250740489207306901, 7.61668218273475816045616360166, 7.75561733435882782922605833583, 7.925541553474182287446286121927, 8.093569712669832246212012715761, 8.656173720538163881522305901026, 8.804250593409703629129334147213, 9.283874765210793305804957032726, 9.551534676747281942827720215751
