| L(s) = 1 | + 2·2-s + 4-s − 2·5-s + 10·7-s + 2·8-s − 4·10-s + 4·13-s + 20·14-s + 4·17-s − 2·20-s + 4·23-s + 25-s + 8·26-s + 10·28-s + 2·29-s − 14·32-s + 8·34-s − 20·35-s − 4·40-s + 6·41-s + 6·43-s + 8·46-s + 4·47-s + 61·49-s + 2·50-s + 4·52-s + 8·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s + 3.77·7-s + 0.707·8-s − 1.26·10-s + 1.10·13-s + 5.34·14-s + 0.970·17-s − 0.447·20-s + 0.834·23-s + 1/5·25-s + 1.56·26-s + 1.88·28-s + 0.371·29-s − 2.47·32-s + 1.37·34-s − 3.38·35-s − 0.632·40-s + 0.937·41-s + 0.914·43-s + 1.17·46-s + 0.583·47-s + 61/7·49-s + 0.282·50-s + 0.554·52-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.59185065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.59185065\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) | |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) | |
| good | 2 | $D_{4}$ | \( ( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) | 4.2.ac_d_ag_n |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_aw_a_nz |
| 13 | $D_4\times C_2$ | \( 1 - 4 T - T^{2} + 36 T^{3} - 88 T^{4} + 36 p T^{5} - p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.ae_ab_bk_adk |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 9 T^{2} + 36 T^{3} + 64 T^{4} + 36 p T^{5} - 9 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ae_aj_bk_cm |
| 19 | $C_2^3$ | \( 1 - 25 T^{2} + 264 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_az_a_ke |
| 23 | $D_4\times C_2$ | \( 1 - 4 T + 18 T^{2} + 192 T^{3} - 893 T^{4} + 192 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ae_s_hk_abij |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 102 T^{3} - 908 T^{4} + 102 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.ac_ad_dy_abiy |
| 31 | $C_2^2$ | \( ( 1 + 49 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_du_a_gkh |
| 37 | $C_2^3$ | \( 1 - 61 T^{2} + 2352 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_acj_a_dmm |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ag_acd_acc_hpc |
| 43 | $D_4\times C_2$ | \( 1 - 6 T - 7 T^{2} + 6 p T^{3} - 36 p T^{4} + 6 p^{2} T^{5} - 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ag_ah_jy_acho |
| 47 | $D_{4}$ | \( ( 1 - 2 T + 43 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.ae_dm_anw_jvn |
| 53 | $D_4\times C_2$ | \( 1 - 8 T - 45 T^{2} - 24 T^{3} + 5680 T^{4} - 24 p T^{5} - 45 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.ai_abt_ay_ikm |
| 59 | $D_{4}$ | \( ( 1 - 16 T + 169 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.abg_ww_akuq_dtgd |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 79 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.m_hm_cmq_batf |
| 67 | $D_{4}$ | \( ( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.m_ik_cuy_bgrj |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2^3$ | \( 1 - 133 T^{2} + 12360 T^{4} - 133 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_afd_a_shk |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ai_ec_abme_zff |
| 83 | $D_4\times C_2$ | \( 1 + 14 T + 33 T^{2} - 42 T^{3} + 3412 T^{4} - 42 p T^{5} + 33 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.o_bh_abq_fbg |
| 89 | $C_2^2$ | \( ( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ag_afv_acc_bjhc |
| 97 | $D_4\times C_2$ | \( 1 + 20 T + 119 T^{2} + 1740 T^{3} + 30752 T^{4} + 1740 p T^{5} + 119 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.u_ep_coy_btmu |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12220289869536192392951875056, −7.07922228255867580974048846011, −6.96822487245706715784874073098, −6.59619083003170993684471093625, −6.11940009912797954871097613707, −5.70398876400342840188166896146, −5.60771364367026250181033776148, −5.50777585508412383880860670494, −5.47770600967429596645966108099, −4.90651929092983531992339407109, −4.66767428201198465316436183851, −4.64699471999504800250622068115, −4.61878652228554118644340545506, −4.11906200029257255032542827611, −3.84432324992177264791799224351, −3.83474296628085973128220047297, −3.80457199620588840079074655162, −2.91150684348928563099585030446, −2.89150975248740484082675263153, −2.38701966237602684065644491903, −1.98755343968423916475175928881, −1.87684909607700887726205350555, −1.39472164048066488268924076663, −1.00103948950117097142331637699, −0.841089487502002026679821142282,
0.841089487502002026679821142282, 1.00103948950117097142331637699, 1.39472164048066488268924076663, 1.87684909607700887726205350555, 1.98755343968423916475175928881, 2.38701966237602684065644491903, 2.89150975248740484082675263153, 2.91150684348928563099585030446, 3.80457199620588840079074655162, 3.83474296628085973128220047297, 3.84432324992177264791799224351, 4.11906200029257255032542827611, 4.61878652228554118644340545506, 4.64699471999504800250622068115, 4.66767428201198465316436183851, 4.90651929092983531992339407109, 5.47770600967429596645966108099, 5.50777585508412383880860670494, 5.60771364367026250181033776148, 5.70398876400342840188166896146, 6.11940009912797954871097613707, 6.59619083003170993684471093625, 6.96822487245706715784874073098, 7.07922228255867580974048846011, 7.12220289869536192392951875056
