| L(s) = 1 | − 5·5-s − 3·7-s − 5·9-s + 2·11-s + 3·13-s + 11·19-s + 23-s + 5·25-s − 2·27-s + 29-s + 31-s + 15·35-s − 18·37-s + 2·41-s − 3·43-s + 25·45-s − 7·47-s + 6·49-s + 5·53-s − 10·55-s + 6·59-s − 14·61-s + 15·63-s − 15·65-s − 6·67-s − 30·71-s − 25·73-s + ⋯ |
| L(s) = 1 | − 2.23·5-s − 1.13·7-s − 5/3·9-s + 0.603·11-s + 0.832·13-s + 2.52·19-s + 0.208·23-s + 25-s − 0.384·27-s + 0.185·29-s + 0.179·31-s + 2.53·35-s − 2.95·37-s + 0.312·41-s − 0.457·43-s + 3.72·45-s − 1.02·47-s + 6/7·49-s + 0.686·53-s − 1.34·55-s + 0.781·59-s − 1.79·61-s + 1.88·63-s − 1.86·65-s − 0.733·67-s − 3.56·71-s − 2.92·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 7^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 2 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.3.a_f_c |
| 5 | $S_4\times C_2$ | \( 1 + p T + 4 p T^{2} + 49 T^{3} + 4 p^{2} T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.5.f_u_bx |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 31 T^{2} - 42 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_bf_abq |
| 17 | $S_4\times C_2$ | \( 1 + 15 T^{2} + 54 T^{3} + 15 p T^{4} + p^{3} T^{6} \) | 3.17.a_p_cc |
| 19 | $S_4\times C_2$ | \( 1 - 11 T + 82 T^{2} - 431 T^{3} + 82 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.al_de_aqp |
| 23 | $S_4\times C_2$ | \( 1 - T + 16 T^{2} - 33 T^{3} + 16 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ab_q_abh |
| 29 | $S_4\times C_2$ | \( 1 - T + 66 T^{2} - 45 T^{3} + 66 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ab_co_abt |
| 31 | $S_4\times C_2$ | \( 1 - T + 80 T^{2} - 39 T^{3} + 80 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ab_dc_abn |
| 37 | $S_4\times C_2$ | \( 1 + 18 T + 5 p T^{2} + 1282 T^{3} + 5 p^{2} T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.s_hd_bxi |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 87 T^{2} - 60 T^{3} + 87 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ac_dj_aci |
| 43 | $S_4\times C_2$ | \( 1 + 3 T + 92 T^{2} + 295 T^{3} + 92 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.d_do_lj |
| 47 | $S_4\times C_2$ | \( 1 + 7 T + 98 T^{2} + 359 T^{3} + 98 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.h_du_nv |
| 53 | $S_4\times C_2$ | \( 1 - 5 T + 14 T^{2} + 71 T^{3} + 14 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.af_o_ct |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 89 T^{2} - 784 T^{3} + 89 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_dl_abee |
| 61 | $S_4\times C_2$ | \( 1 + 14 T + 115 T^{2} + 596 T^{3} + 115 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.o_el_wy |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 44 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_cz_bs |
| 71 | $S_4\times C_2$ | \( 1 + 30 T + 455 T^{2} + 4510 T^{3} + 455 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.be_rn_grm |
| 73 | $S_4\times C_2$ | \( 1 + 25 T + 406 T^{2} + 4087 T^{3} + 406 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.z_pq_gbf |
| 79 | $S_4\times C_2$ | \( 1 + 11 T + 44 T^{2} - 309 T^{3} + 44 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.l_bs_alx |
| 83 | $S_4\times C_2$ | \( 1 - 11 T + 240 T^{2} - 1593 T^{3} + 240 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.al_jg_acjh |
| 89 | $S_4\times C_2$ | \( 1 + 5 T - 26 T^{2} - 1557 T^{3} - 26 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.f_aba_achx |
| 97 | $S_4\times C_2$ | \( 1 + 41 T + 830 T^{2} + 10179 T^{3} + 830 p T^{4} + 41 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.bp_bfy_pbn |
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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
−8.325477499025564382141663059820, −7.81667203913267345872150835996, −7.70817050390943429456443027838, −7.33327532107313027464264473861, −7.12865742016169870189413716170, −6.95906886405662275878052347044, −6.89524490414698543057474606115, −6.22950893870125877883512622429, −6.04742362779748409472997693034, −5.92514649045053920091058487087, −5.49139911023425415249002691948, −5.37016945321856961203457417348, −5.18974188754184674804491676630, −4.53041770171701140854311951174, −4.41727607180983923656695429113, −4.03158699384157137891715359924, −3.76383755189938565888623262214, −3.55943697210016536048377257086, −3.33224707325105503672970568834, −2.97578845893840946747609977342, −2.81772484306215453639203965372, −2.67883229965991069133543852661, −1.56856185535403682286814222706, −1.53377698942104893790111030014, −1.10010702211035994583084166248, 0, 0, 0,
1.10010702211035994583084166248, 1.53377698942104893790111030014, 1.56856185535403682286814222706, 2.67883229965991069133543852661, 2.81772484306215453639203965372, 2.97578845893840946747609977342, 3.33224707325105503672970568834, 3.55943697210016536048377257086, 3.76383755189938565888623262214, 4.03158699384157137891715359924, 4.41727607180983923656695429113, 4.53041770171701140854311951174, 5.18974188754184674804491676630, 5.37016945321856961203457417348, 5.49139911023425415249002691948, 5.92514649045053920091058487087, 6.04742362779748409472997693034, 6.22950893870125877883512622429, 6.89524490414698543057474606115, 6.95906886405662275878052347044, 7.12865742016169870189413716170, 7.33327532107313027464264473861, 7.70817050390943429456443027838, 7.81667203913267345872150835996, 8.325477499025564382141663059820
