| L(s) = 1 | − 3·4-s + 3·5-s − 3·7-s − 8-s + 3·11-s + 3·13-s + 3·16-s + 9·17-s − 9·20-s − 6·23-s − 6·25-s + 9·28-s − 9·29-s + 24·31-s + 6·32-s − 9·35-s + 6·37-s − 3·40-s + 6·41-s − 21·43-s − 9·44-s + 3·47-s − 12·49-s − 9·52-s + 18·53-s + 9·55-s + 3·56-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.34·5-s − 1.13·7-s − 0.353·8-s + 0.904·11-s + 0.832·13-s + 3/4·16-s + 2.18·17-s − 2.01·20-s − 1.25·23-s − 6/5·25-s + 1.70·28-s − 1.67·29-s + 4.31·31-s + 1.06·32-s − 1.52·35-s + 0.986·37-s − 0.474·40-s + 0.937·41-s − 3.20·43-s − 1.35·44-s + 0.437·47-s − 1.71·49-s − 1.24·52-s + 2.47·53-s + 1.21·55-s + 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.248953208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.248953208\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 19 | | \( 1 \) | |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T^{2} + T^{3} + 3 p T^{4} + p^{3} T^{6} \) | 3.2.a_d_b |
| 5 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 27 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ad_p_abb |
| 7 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 41 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.d_v_bp |
| 11 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 29 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ad_p_abd |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 30 T^{2} - 59 T^{3} + 30 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_be_ach |
| 17 | $A_4\times C_2$ | \( 1 - 9 T + 57 T^{2} - 253 T^{3} + 57 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.aj_cf_ajt |
| 23 | $A_4\times C_2$ | \( 1 + 6 T + 54 T^{2} + 203 T^{3} + 54 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.g_cc_hv |
| 29 | $A_4\times C_2$ | \( 1 + 9 T + 66 T^{2} + 469 T^{3} + 66 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.j_co_sb |
| 31 | $A_4\times C_2$ | \( 1 - 24 T + 282 T^{2} - 1977 T^{3} + 282 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ay_kw_acyb |
| 37 | $A_4\times C_2$ | \( 1 - 6 T + 102 T^{2} - 393 T^{3} + 102 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_dy_apd |
| 41 | $A_4\times C_2$ | \( 1 - 6 T + 96 T^{2} - 441 T^{3} + 96 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ag_ds_aqz |
| 43 | $A_4\times C_2$ | \( 1 + 21 T + 240 T^{2} + 1825 T^{3} + 240 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.v_jg_csf |
| 47 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 495 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ad_bh_atb |
| 53 | $A_4\times C_2$ | \( 1 - 18 T + 210 T^{2} - 1619 T^{3} + 210 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.as_ic_ackh |
| 59 | $A_4\times C_2$ | \( 1 - 15 T + 243 T^{2} - 1859 T^{3} + 243 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ap_jj_actn |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 1135 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.j_en_brr |
| 67 | $A_4\times C_2$ | \( 1 + 6 T + 129 T^{2} + 508 T^{3} + 129 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_ez_to |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 204 T^{2} - 1125 T^{3} + 204 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_hw_abrh |
| 73 | $A_4\times C_2$ | \( 1 + 6 T + 168 T^{2} + 767 T^{3} + 168 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.g_gm_bdn |
| 79 | $A_4\times C_2$ | \( 1 - 9 T + 216 T^{2} - 1369 T^{3} + 216 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.aj_ii_acar |
| 83 | $A_4\times C_2$ | \( 1 - 15 T + 285 T^{2} - 2439 T^{3} + 285 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ap_kz_adpv |
| 89 | $A_4\times C_2$ | \( 1 + 42 T^{2} + 1125 T^{3} + 42 p T^{4} + p^{3} T^{6} \) | 3.89.a_bq_brh |
| 97 | $A_4\times C_2$ | \( 1 + 6 T + 246 T^{2} + 895 T^{3} + 246 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.g_jm_bil |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.953449545615640000100835189648, −7.37543056755714803278995892556, −7.23789188937206170387185495946, −6.64403946817044281969703312610, −6.60042384093963534853645336617, −6.21637233083310340510105185052, −6.12585299449138411813521988457, −5.90722834186597376930563851335, −5.77283443140701463812373851972, −5.47991933697402943491407553552, −5.03598891302171777081717378595, −4.75846680606184486605457457531, −4.65464582139483557256417580821, −4.12961837170222476146029787514, −3.89008569540406338779097759464, −3.80270961804269768889853734081, −3.24953907009413870966778340982, −3.21557383356775530205009893249, −2.93978745038615739041312828548, −2.18684791128046091260184426860, −2.11402975531072306752974516724, −1.70998649748166123457238547730, −1.14984409452821762277715860013, −0.71400243975855102173931326550, −0.52372169175375729920119235283,
0.52372169175375729920119235283, 0.71400243975855102173931326550, 1.14984409452821762277715860013, 1.70998649748166123457238547730, 2.11402975531072306752974516724, 2.18684791128046091260184426860, 2.93978745038615739041312828548, 3.21557383356775530205009893249, 3.24953907009413870966778340982, 3.80270961804269768889853734081, 3.89008569540406338779097759464, 4.12961837170222476146029787514, 4.65464582139483557256417580821, 4.75846680606184486605457457531, 5.03598891302171777081717378595, 5.47991933697402943491407553552, 5.77283443140701463812373851972, 5.90722834186597376930563851335, 6.12585299449138411813521988457, 6.21637233083310340510105185052, 6.60042384093963534853645336617, 6.64403946817044281969703312610, 7.23789188937206170387185495946, 7.37543056755714803278995892556, 7.953449545615640000100835189648
