| L(s) = 1 | − 2·4-s − 4·5-s − 2·16-s − 12·19-s + 8·20-s + 10·25-s + 12·29-s − 12·31-s − 24·37-s + 12·41-s + 16·43-s + 24·47-s − 16·49-s − 12·59-s − 32·61-s + 8·64-s + 12·67-s − 12·71-s − 24·73-s + 24·76-s − 8·79-s + 8·80-s + 12·89-s + 48·95-s − 36·97-s − 20·100-s + 24·101-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s − 1/2·16-s − 2.75·19-s + 1.78·20-s + 2·25-s + 2.22·29-s − 2.15·31-s − 3.94·37-s + 1.87·41-s + 2.43·43-s + 3.50·47-s − 2.28·49-s − 1.56·59-s − 4.09·61-s + 64-s + 1.46·67-s − 1.42·71-s − 2.80·73-s + 2.75·76-s − 0.900·79-s + 0.894·80-s + 1.27·89-s + 4.92·95-s − 3.65·97-s − 2·100-s + 2.38·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 13 | | \( 1 \) | |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + p T^{2} + 3 p T^{4} + p^{3} T^{6} + p^{4} T^{8} \) | 4.2.a_c_a_g |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T^{2} + 12 T^{3} + 123 T^{4} + 12 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_q_m_et |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 32 T^{2} + 486 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_bg_a_ss |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 50 T^{2} + 1176 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_by_a_btg |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 4 p T^{2} + 372 T^{3} + 1722 T^{4} + 372 p T^{5} + 4 p^{3} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.m_cy_oi_cog |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 32 T^{2} - 144 T^{3} + 342 T^{4} - 144 p T^{5} + 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bg_afo_ne |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 146 T^{2} - 972 T^{3} + 6552 T^{4} - 972 p T^{5} + 146 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.am_fq_ablk_jsa |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 130 T^{2} + 936 T^{3} + 6171 T^{4} + 936 p T^{5} + 130 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.m_fa_bka_jdj |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 340 T^{2} + 3240 T^{3} + 22950 T^{4} + 3240 p T^{5} + 340 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.y_nc_euq_bhys |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 158 T^{2} - 1260 T^{3} + 9816 T^{4} - 1260 p T^{5} + 158 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.am_gc_abwm_ono |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 208 T^{2} - 1660 T^{3} + 12667 T^{4} - 1660 p T^{5} + 208 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.aq_ia_aclw_stf |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 24 T + 386 T^{2} - 4032 T^{3} + 32520 T^{4} - 4032 p T^{5} + 386 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ay_ow_afzc_bwcu |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 68 T^{2} + 5046 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_cq_a_hmc |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 182 T^{2} + 1260 T^{3} + 12408 T^{4} + 1260 p T^{5} + 182 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.m_ha_bwm_sjg |
| 61 | $D_{4}$ | \( ( 1 + 16 T + 3 p T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bg_xy_loi_ectn |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 208 T^{2} - 1980 T^{3} + 19875 T^{4} - 1980 p T^{5} + 208 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.am_ia_acye_bdkl |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 170 T^{2} + 1476 T^{3} + 12912 T^{4} + 1476 p T^{5} + 170 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.m_go_ceu_tcq |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 24 T + 448 T^{2} + 5292 T^{3} + 53571 T^{4} + 5292 p T^{5} + 448 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.y_rg_hvo_dbgl |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 214 T^{2} + 992 T^{3} + 19627 T^{4} + 992 p T^{5} + 214 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.i_ig_bme_bdax |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 200 T^{2} + 23478 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_hs_a_bita |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 242 T^{2} - 2484 T^{3} + 27168 T^{4} - 2484 p T^{5} + 242 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.am_ji_adro_boey |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 36 T + 856 T^{2} + 13068 T^{3} + 152355 T^{4} + 13068 p T^{5} + 856 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.bk_bgy_tiq_irjv |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.95455193576099243141917677583, −5.90764912928292678872710528414, −5.36607640279771164103047389373, −5.34982996857499241834664722974, −5.06811527816415084595726110739, −4.79831511428635743703881184193, −4.62702560185648459182675907361, −4.55097047358203881129619948739, −4.52119066048708405399259247721, −4.16427548304270471331214686745, −4.03564107565263184448862799046, −3.95610129861326889772558763160, −3.83660871330002462382940524088, −3.41899446381875465190826166948, −3.23369821692786009585882604563, −3.17423251250904959827990384390, −2.81930099304209459765253704309, −2.55179042119121867170438308940, −2.50703188017614302076516306832, −2.15845295610783386502321583769, −1.90191016461503590174130563520, −1.65848245701307417889341431677, −1.27932398946133373027220268351, −1.13030520102778794161854977420, −0.816245389920147711546190153018, 0, 0, 0, 0,
0.816245389920147711546190153018, 1.13030520102778794161854977420, 1.27932398946133373027220268351, 1.65848245701307417889341431677, 1.90191016461503590174130563520, 2.15845295610783386502321583769, 2.50703188017614302076516306832, 2.55179042119121867170438308940, 2.81930099304209459765253704309, 3.17423251250904959827990384390, 3.23369821692786009585882604563, 3.41899446381875465190826166948, 3.83660871330002462382940524088, 3.95610129861326889772558763160, 4.03564107565263184448862799046, 4.16427548304270471331214686745, 4.52119066048708405399259247721, 4.55097047358203881129619948739, 4.62702560185648459182675907361, 4.79831511428635743703881184193, 5.06811527816415084595726110739, 5.34982996857499241834664722974, 5.36607640279771164103047389373, 5.90764912928292678872710528414, 5.95455193576099243141917677583
