| L(s) = 1 | − 3·3-s + 3·5-s + 3·7-s + 2·8-s + 6·9-s − 9·15-s + 12·19-s − 9·21-s − 6·24-s + 6·25-s − 10·27-s + 6·29-s + 3·31-s + 9·35-s + 6·37-s + 6·40-s + 9·43-s + 18·45-s − 12·47-s − 9·49-s + 6·56-s − 36·57-s − 6·59-s − 3·61-s + 18·63-s − 4·64-s + 9·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 1.13·7-s + 0.707·8-s + 2·9-s − 2.32·15-s + 2.75·19-s − 1.96·21-s − 1.22·24-s + 6/5·25-s − 1.92·27-s + 1.11·29-s + 0.538·31-s + 1.52·35-s + 0.986·37-s + 0.948·40-s + 1.37·43-s + 2.68·45-s − 1.75·47-s − 9/7·49-s + 0.801·56-s − 4.76·57-s − 0.781·59-s − 0.384·61-s + 2.26·63-s − 1/2·64-s + 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.926551099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.926551099\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 13 | | \( 1 \) | |
| good | 2 | $D_{6}$ | \( 1 - p T^{3} + p^{3} T^{6} \) | 3.2.a_a_ac |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} - 39 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ad_s_abn |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 16 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.11.a_j_q |
| 17 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 98 T^{3} + 9 p T^{4} + p^{3} T^{6} \) | 3.17.a_j_adu |
| 19 | $S_4\times C_2$ | \( 1 - 12 T + 93 T^{2} - 460 T^{3} + 93 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.am_dp_ars |
| 23 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 96 T^{3} + 21 p T^{4} + p^{3} T^{6} \) | 3.23.a_v_ds |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 334 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ag_dj_amw |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 177 T^{3} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_a_gv |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 75 T^{2} - 452 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_cx_ark |
| 41 | $S_4\times C_2$ | \( 1 + 99 T^{2} + 26 T^{3} + 99 p T^{4} + p^{3} T^{6} \) | 3.41.a_dv_ba |
| 43 | $S_4\times C_2$ | \( 1 - 9 T + 144 T^{2} - 763 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aj_fo_abdj |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 135 T^{2} + 922 T^{3} + 135 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.m_ff_bjm |
| 53 | $S_4\times C_2$ | \( 1 + 135 T^{2} + 16 T^{3} + 135 p T^{4} + p^{3} T^{6} \) | 3.53.a_ff_q |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 165 T^{2} + 694 T^{3} + 165 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.g_gj_bas |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 162 T^{2} + 299 T^{3} + 162 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.d_gg_ln |
| 67 | $S_4\times C_2$ | \( 1 - 9 T + 120 T^{2} - 855 T^{3} + 120 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.aj_eq_abgx |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 165 T^{2} + 1022 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.m_gj_bni |
| 73 | $S_4\times C_2$ | \( 1 - 21 T + 312 T^{2} - 2977 T^{3} + 312 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.av_ma_aekn |
| 79 | $S_4\times C_2$ | \( 1 + 3 T + 144 T^{2} + 111 T^{3} + 144 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_fo_eh |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 309 T^{2} + 2820 T^{3} + 309 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.s_lx_eem |
| 89 | $S_4\times C_2$ | \( 1 - 30 T + 555 T^{2} - 6222 T^{3} + 555 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.abe_vj_ajfi |
| 97 | $S_4\times C_2$ | \( 1 - 15 T + 282 T^{2} - 2321 T^{3} + 282 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ap_kw_adlh |
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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
−7.73242805947301398077101233380, −7.67001967339431837056579034132, −7.38374377854596774315812180517, −7.23828357222469569841866153965, −6.61671312441268628995401131953, −6.40733717475707245380090630623, −6.34397721532183717445105786044, −6.22852068467580380019077221423, −5.71833272507841333575111256881, −5.44869910711496184286412109401, −5.12268696289500800055779870207, −5.08020820731593002369805513109, −4.94906115153582998029878879676, −4.59790466331063369792514803537, −4.20492491555449409220604760591, −4.08842349534311842685138679788, −3.34536915768417045049606506695, −3.22764290651301528845936300125, −2.79777535135361529284954269599, −2.44506953035160570273660974567, −1.81423992247569161045157607596, −1.62427717701669106561410556519, −1.39857396900715405153260297307, −0.76389919425840720076528653597, −0.74712520427779293049359071043,
0.74712520427779293049359071043, 0.76389919425840720076528653597, 1.39857396900715405153260297307, 1.62427717701669106561410556519, 1.81423992247569161045157607596, 2.44506953035160570273660974567, 2.79777535135361529284954269599, 3.22764290651301528845936300125, 3.34536915768417045049606506695, 4.08842349534311842685138679788, 4.20492491555449409220604760591, 4.59790466331063369792514803537, 4.94906115153582998029878879676, 5.08020820731593002369805513109, 5.12268696289500800055779870207, 5.44869910711496184286412109401, 5.71833272507841333575111256881, 6.22852068467580380019077221423, 6.34397721532183717445105786044, 6.40733717475707245380090630623, 6.61671312441268628995401131953, 7.23828357222469569841866153965, 7.38374377854596774315812180517, 7.67001967339431837056579034132, 7.73242805947301398077101233380
