| L(s) = 1 | + 4·5-s + 6·9-s − 6·13-s + 2·17-s + 14·25-s − 10·29-s − 2·37-s + 10·41-s + 24·45-s + 14·49-s − 28·53-s − 10·61-s − 24·65-s + 12·73-s + 9·81-s + 8·85-s − 20·89-s − 36·97-s − 2·101-s + 24·109-s − 14·113-s − 36·117-s + 22·121-s + 64·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2·9-s − 1.66·13-s + 0.485·17-s + 14/5·25-s − 1.85·29-s − 0.328·37-s + 1.56·41-s + 3.57·45-s + 2·49-s − 3.84·53-s − 1.28·61-s − 2.97·65-s + 1.40·73-s + 81-s + 0.867·85-s − 2.11·89-s − 3.65·97-s − 0.199·101-s + 2.29·109-s − 1.31·113-s − 3.32·117-s + 2·121-s + 5.72·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.636013848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.636013848\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) | 4.3.a_ag_a_bb |
| 5 | $C_2^2$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ae_c_aq_dn |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_ao_a_fr |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_aw_a_nz |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) | 4.17.ac_r_dq_ahg |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abm_a_bpr |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_abu_a_cjb |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) | 4.29.k_bd_fa_bya |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.31.a_eu_a_inu |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) | 4.37.c_bl_aig_aqm |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) | 4.41.ak_bp_iw_adkm |
| 43 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_adi_a_ifj |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.47.a_hg_a_tpu |
| 53 | $C_2^2$ | \( ( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.bc_so_idc_crhr |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aeo_a_plr |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) | 4.61.k_cj_abfy_amhg |
| 67 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_afe_a_txz |
| 71 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_afm_a_wjr |
| 73 | $C_2^2$ | \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.am_abm_aqq_zot |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.83.a_mu_a_cjdu |
| 89 | $C_2^2$ | \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.u_es_cyy_bxyp |
| 97 | $C_2^2$ | \( ( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.bk_bdy_rgq_hpbf |
| 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
−7.975445093240735173481378788063, −7.76117247492538438037530362028, −7.53790765312086117280878972105, −7.29225986188843574284946001380, −7.26904459171443857326305880975, −6.93822264004032859652818746512, −6.56546368161579126123375522011, −6.36346122064774768861193478682, −6.26975979927335576912515009282, −5.67140035079707015491724778430, −5.62213449270403166404704212725, −5.34646869418283280224301128829, −5.11063934362423115047210656723, −4.75217347570088163709179678261, −4.39491055892099830466380941802, −4.37047038976356805569766688107, −3.94482325744288031830849481082, −3.59302494709322558959505063756, −2.99872450239467610938363441290, −2.73588033518958669744182930141, −2.64965181453340557889179221113, −1.79290543314879756681242465962, −1.76216394319035606246726551869, −1.55645804680609255556160859075, −0.70193976957818435534057543417,
0.70193976957818435534057543417, 1.55645804680609255556160859075, 1.76216394319035606246726551869, 1.79290543314879756681242465962, 2.64965181453340557889179221113, 2.73588033518958669744182930141, 2.99872450239467610938363441290, 3.59302494709322558959505063756, 3.94482325744288031830849481082, 4.37047038976356805569766688107, 4.39491055892099830466380941802, 4.75217347570088163709179678261, 5.11063934362423115047210656723, 5.34646869418283280224301128829, 5.62213449270403166404704212725, 5.67140035079707015491724778430, 6.26975979927335576912515009282, 6.36346122064774768861193478682, 6.56546368161579126123375522011, 6.93822264004032859652818746512, 7.26904459171443857326305880975, 7.29225986188843574284946001380, 7.53790765312086117280878972105, 7.76117247492538438037530362028, 7.975445093240735173481378788063
