| L(s) = 1 | + 2·3-s + 4·7-s − 2·9-s − 11-s − 3·13-s − 14·17-s − 3·19-s + 8·21-s + 8·23-s − 9·27-s − 5·29-s + 31-s − 2·33-s − 5·37-s − 6·39-s + 41-s − 5·43-s + 9·47-s − 6·49-s − 28·51-s − 31·53-s − 6·57-s + 6·59-s + 3·61-s − 8·63-s − 13·67-s + 16·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s − 2/3·9-s − 0.301·11-s − 0.832·13-s − 3.39·17-s − 0.688·19-s + 1.74·21-s + 1.66·23-s − 1.73·27-s − 0.928·29-s + 0.179·31-s − 0.348·33-s − 0.821·37-s − 0.960·39-s + 0.156·41-s − 0.762·43-s + 1.31·47-s − 6/7·49-s − 3.92·51-s − 4.25·53-s − 0.794·57-s + 0.781·59-s + 0.384·61-s − 1.00·63-s − 1.58·67-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 7 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_g_ah |
| 7 | $A_4\times C_2$ | \( 1 - 4 T + 22 T^{2} - 55 T^{3} + 22 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_w_acd |
| 11 | $A_4\times C_2$ | \( 1 + T + 29 T^{2} + 23 T^{3} + 29 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.11.b_bd_x |
| 13 | $A_4\times C_2$ | \( 1 + 3 T + 3 T^{2} - 25 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.d_d_az |
| 17 | $A_4\times C_2$ | \( 1 + 14 T + 112 T^{2} + 555 T^{3} + 112 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.o_ei_vj |
| 23 | $A_4\times C_2$ | \( 1 - 8 T + 60 T^{2} - 243 T^{3} + 60 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ai_ci_ajj |
| 29 | $A_4\times C_2$ | \( 1 + 5 T + 91 T^{2} + 285 T^{3} + 91 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.f_dn_kz |
| 31 | $A_4\times C_2$ | \( 1 - T + 63 T^{2} - 115 T^{3} + 63 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ab_cl_ael |
| 37 | $A_4\times C_2$ | \( 1 + 5 T + p T^{2} - 25 T^{3} + p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.f_bl_az |
| 41 | $A_4\times C_2$ | \( 1 - T + p T^{2} + 73 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ab_bp_cv |
| 43 | $A_4\times C_2$ | \( 1 + 5 T + 16 T^{2} + 113 T^{3} + 16 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.f_q_ej |
| 47 | $A_4\times C_2$ | \( 1 - 9 T + 77 T^{2} - 535 T^{3} + 77 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.aj_cz_aup |
| 53 | $A_4\times C_2$ | \( 1 + 31 T + 462 T^{2} + 4191 T^{3} + 462 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.bf_ru_gff |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 508 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_fh_ato |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 173 T^{2} - 367 T^{3} + 173 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ad_gr_aod |
| 67 | $A_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 1573 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.n_ht_cin |
| 71 | $A_4\times C_2$ | \( 1 + 7 T + 212 T^{2} + 947 T^{3} + 212 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.h_ie_bkl |
| 73 | $A_4\times C_2$ | \( 1 + T + 189 T^{2} + 199 T^{3} + 189 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.73.b_hh_hr |
| 79 | $A_4\times C_2$ | \( 1 - 18 T + 254 T^{2} - 31 p T^{3} + 254 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.as_ju_adqf |
| 83 | $A_4\times C_2$ | \( 1 - 3 T - 21 T^{2} - 629 T^{3} - 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ad_av_ayf |
| 89 | $A_4\times C_2$ | \( 1 + 20 T + 318 T^{2} + 3435 T^{3} + 318 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.u_mg_fcd |
| 97 | $A_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 2353 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.an_lf_admn |
| 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
−7.59560037263763430453614314225, −6.89244690367551691333498493110, −6.81494020797618312300308720666, −6.81318119746270963512370600958, −6.50499032980325763257116174955, −6.07541162281930594941912676119, −6.06403308641084553298425315647, −5.61000459636287331195023958020, −5.21096926983598594365003304519, −5.17751350002026515101751875112, −4.88869648004747131139606027918, −4.62572684522604573198777321625, −4.49482366776071365596298165351, −4.38374489721812664341607923201, −3.84035847487999676192514607206, −3.71998442205173957227443118014, −3.19106099124531897616364711897, −3.11632180715378391869153508418, −2.84599649368881805744741691858, −2.32595351100790542565977182393, −2.30441746569013997102147899244, −2.20923838849575574211228593640, −1.63657300475069191561447173639, −1.51671879359981659705205137181, −1.10206462875254636013754300128, 0, 0, 0,
1.10206462875254636013754300128, 1.51671879359981659705205137181, 1.63657300475069191561447173639, 2.20923838849575574211228593640, 2.30441746569013997102147899244, 2.32595351100790542565977182393, 2.84599649368881805744741691858, 3.11632180715378391869153508418, 3.19106099124531897616364711897, 3.71998442205173957227443118014, 3.84035847487999676192514607206, 4.38374489721812664341607923201, 4.49482366776071365596298165351, 4.62572684522604573198777321625, 4.88869648004747131139606027918, 5.17751350002026515101751875112, 5.21096926983598594365003304519, 5.61000459636287331195023958020, 6.06403308641084553298425315647, 6.07541162281930594941912676119, 6.50499032980325763257116174955, 6.81318119746270963512370600958, 6.81494020797618312300308720666, 6.89244690367551691333498493110, 7.59560037263763430453614314225
