| L(s) = 1 | + 6·5-s + 3·7-s − 3·11-s − 3·13-s + 9·17-s + 3·19-s − 6·23-s + 12·25-s + 12·29-s + 12·31-s + 18·35-s − 3·37-s − 3·41-s + 12·43-s + 6·47-s − 6·49-s + 18·53-s − 18·55-s + 21·59-s + 6·61-s − 18·65-s − 6·67-s + 9·71-s + 6·73-s − 9·77-s − 6·79-s + 6·83-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 1.13·7-s − 0.904·11-s − 0.832·13-s + 2.18·17-s + 0.688·19-s − 1.25·23-s + 12/5·25-s + 2.22·29-s + 2.15·31-s + 3.04·35-s − 0.493·37-s − 0.468·41-s + 1.82·43-s + 0.875·47-s − 6/7·49-s + 2.47·53-s − 2.42·55-s + 2.73·59-s + 0.768·61-s − 2.23·65-s − 0.733·67-s + 1.06·71-s + 0.702·73-s − 1.02·77-s − 0.675·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.58061589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.58061589\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 5 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 63 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ag_y_acl |
| 7 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 25 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ad_p_az |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 63 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.d_p_cl |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 61 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.d_bh_cj |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) | 3.17.aj_da_amv |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 115 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ad_bh_ael |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 225 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_ci_ir |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 114 T^{2} - 639 T^{3} + 114 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.am_ek_ayp |
| 31 | $A_4\times C_2$ | \( 1 - 12 T + 132 T^{2} - 763 T^{3} + 132 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.am_fc_abdj |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 223 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.d_dj_ip |
| 41 | $A_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 27 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.d_cr_bb |
| 43 | $A_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1051 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.am_gm_abol |
| 47 | $A_4\times C_2$ | \( 1 - 6 T + 78 T^{2} - 297 T^{3} + 78 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ag_da_all |
| 53 | $A_4\times C_2$ | \( 1 - 18 T + 240 T^{2} - 1989 T^{3} + 240 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.as_jg_acyn |
| 59 | $A_4\times C_2$ | \( 1 - 21 T + 321 T^{2} - 2799 T^{3} + 321 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.av_mj_aedr |
| 61 | $A_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 785 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ag_fc_abef |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 695 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_fu_bat |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 51 T^{2} - 279 T^{3} + 51 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_bz_akt |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 479 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ag_fu_asl |
| 79 | $A_4\times C_2$ | \( 1 + 6 T + 186 T^{2} + 1001 T^{3} + 186 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.g_he_bmn |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 222 T^{2} - 945 T^{3} + 222 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ag_io_abkj |
| 89 | $A_4\times C_2$ | \( 1 + 78 T^{2} + 999 T^{3} + 78 p T^{4} + p^{3} T^{6} \) | 3.89.a_da_bml |
| 97 | $A_4\times C_2$ | \( 1 - 15 T + 222 T^{2} - 2891 T^{3} + 222 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ap_io_aehf |
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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
−7.53202984832275917344208065214, −7.21891217384001208535665003707, −7.14974201982185835261954690933, −6.60817524694796281935181237387, −6.41685150311298921031369456905, −6.13589881531786937928563170098, −6.08861581783971454594087677367, −5.62537675953641429753508240370, −5.38448033786124306784242980145, −5.33718938071608883229952111509, −5.11753069065506214551513803209, −5.00200957024593138258777129831, −4.47564152741265806105628412848, −4.17845183429601549261190604848, −3.89090279473378394021060420899, −3.66271382338925606727635830532, −2.85157331001329287748500645941, −2.84094697524202684181124667023, −2.73049826698470177507556924675, −2.22043797121223692619418085542, −1.91770890624027613017576682947, −1.91458279324289501580709919286, −1.15247881633739083881393121330, −0.975048173104869064912207188932, −0.69012707925880904738331999239,
0.69012707925880904738331999239, 0.975048173104869064912207188932, 1.15247881633739083881393121330, 1.91458279324289501580709919286, 1.91770890624027613017576682947, 2.22043797121223692619418085542, 2.73049826698470177507556924675, 2.84094697524202684181124667023, 2.85157331001329287748500645941, 3.66271382338925606727635830532, 3.89090279473378394021060420899, 4.17845183429601549261190604848, 4.47564152741265806105628412848, 5.00200957024593138258777129831, 5.11753069065506214551513803209, 5.33718938071608883229952111509, 5.38448033786124306784242980145, 5.62537675953641429753508240370, 6.08861581783971454594087677367, 6.13589881531786937928563170098, 6.41685150311298921031369456905, 6.60817524694796281935181237387, 7.14974201982185835261954690933, 7.21891217384001208535665003707, 7.53202984832275917344208065214
