| L(s) = 1 | − 4-s + 4·7-s − 4·13-s + 16-s + 18·19-s − 13·25-s − 4·28-s + 14·31-s + 4·37-s + 10·43-s + 10·49-s + 4·52-s + 24·61-s − 5·64-s − 12·67-s + 22·73-s − 18·76-s + 10·79-s − 16·91-s + 26·97-s + 13·100-s − 8·103-s + 16·109-s + 4·112-s − 20·121-s − 14·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.51·7-s − 1.10·13-s + 1/4·16-s + 4.12·19-s − 2.59·25-s − 0.755·28-s + 2.51·31-s + 0.657·37-s + 1.52·43-s + 10/7·49-s + 0.554·52-s + 3.07·61-s − 5/8·64-s − 1.46·67-s + 2.57·73-s − 2.06·76-s + 1.12·79-s − 1.67·91-s + 2.63·97-s + 1.29·100-s − 0.788·103-s + 1.53·109-s + 0.377·112-s − 1.81·121-s − 1.25·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.343809424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.343809424\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 2 | $D_4$ | \( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} \) | 4.2.a_b_a_a |
| 5 | $D_4\times C_2$ | \( 1 + 13 T^{2} + 84 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_n_a_dg |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_u_a_ne |
| 17 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 846 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_bo_a_bgo |
| 19 | $D_{4}$ | \( ( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.as_gz_abvu_jii |
| 23 | $D_4$ | \( 1 + 13 T^{2} + 696 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_n_a_bau |
| 29 | $D_4\times C_2$ | \( 1 + 73 T^{2} + 2808 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_cv_a_eea |
| 31 | $D_{4}$ | \( ( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.ao_gz_acag_nui |
| 37 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ae_dk_ame_hco |
| 41 | $D_4\times C_2$ | \( 1 + 136 T^{2} + 7854 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_fg_a_lqc |
| 43 | $D_{4}$ | \( ( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ak_cj_axm_jdk |
| 47 | $D_4\times C_2$ | \( 1 + 49 T^{2} + 4812 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_bx_a_hdc |
| 53 | $D_4\times C_2$ | \( 1 + 193 T^{2} + 14856 T^{4} + 193 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_hl_a_vzk |
| 59 | $D_4$ | \( 1 + 208 T^{2} + 17646 T^{4} + 208 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_ia_a_bacs |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.61.ay_rs_ahue_cvyc |
| 67 | $D_{4}$ | \( ( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.m_jw_dds_bmig |
| 71 | $D_4\times C_2$ | \( 1 + 256 T^{2} + 26334 T^{4} + 256 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_jw_a_bmyw |
| 73 | $D_{4}$ | \( ( 1 - 11 T + 168 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.aw_rp_ahvy_dfqy |
| 79 | $D_{4}$ | \( ( 1 - 5 T - 42 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ak_ach_aog_baxw |
| 83 | $D_4\times C_2$ | \( 1 - 127 T^{2} + 11796 T^{4} - 127 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_aex_a_rls |
| 89 | $D_4\times C_2$ | \( 1 + 349 T^{2} + 46284 T^{4} + 349 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_nl_a_cqme |
| 97 | $D_{4}$ | \( ( 1 - 13 T + 228 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.aba_yb_amna_fxge |
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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
−7.50625294430010299885908416150, −7.20283147926650803197942013373, −7.19226838700169198161264631546, −6.45345746824596232353535315252, −6.41865966719493187607115409905, −6.33866200099882094852895118711, −5.73316119342527209367046793357, −5.72976019489439689032558988227, −5.34873402246185813157783487439, −5.20433375031404996931021454062, −4.95925489887696130245337990565, −4.91732876589398239976105394569, −4.63558841708066584945302914228, −4.06324844052153000329251797548, −3.97169503762111151335883298448, −3.83961438666886660148100418234, −3.49895641740204013731155918909, −2.97168808378242862972815889196, −2.75758820417920982148698911055, −2.59025428757784395015559935358, −2.10692426235427251914629193979, −1.83842934541959952707114700357, −1.28002645204984240287764115096, −0.894204021139868532493956564418, −0.69846483658977126747643676616,
0.69846483658977126747643676616, 0.894204021139868532493956564418, 1.28002645204984240287764115096, 1.83842934541959952707114700357, 2.10692426235427251914629193979, 2.59025428757784395015559935358, 2.75758820417920982148698911055, 2.97168808378242862972815889196, 3.49895641740204013731155918909, 3.83961438666886660148100418234, 3.97169503762111151335883298448, 4.06324844052153000329251797548, 4.63558841708066584945302914228, 4.91732876589398239976105394569, 4.95925489887696130245337990565, 5.20433375031404996931021454062, 5.34873402246185813157783487439, 5.72976019489439689032558988227, 5.73316119342527209367046793357, 6.33866200099882094852895118711, 6.41865966719493187607115409905, 6.45345746824596232353535315252, 7.19226838700169198161264631546, 7.20283147926650803197942013373, 7.50625294430010299885908416150
