| L(s) = 1 | − 4·3-s − 8·5-s + 8·7-s + 6·9-s + 12·13-s + 32·15-s + 8·19-s − 32·21-s − 8·23-s + 24·25-s + 4·27-s − 8·31-s − 64·35-s − 48·39-s − 8·41-s − 48·45-s + 16·49-s − 32·57-s − 32·61-s + 48·63-s − 96·65-s − 56·67-s + 32·69-s − 96·75-s − 37·81-s − 40·89-s + 96·91-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.57·5-s + 3.02·7-s + 2·9-s + 3.32·13-s + 8.26·15-s + 1.83·19-s − 6.98·21-s − 1.66·23-s + 24/5·25-s + 0.769·27-s − 1.43·31-s − 10.8·35-s − 7.68·39-s − 1.24·41-s − 7.15·45-s + 16/7·49-s − 4.23·57-s − 4.09·61-s + 6.04·63-s − 11.9·65-s − 6.84·67-s + 3.85·69-s − 11.0·75-s − 4.11·81-s − 4.23·89-s + 10.0·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4297415562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4297415562\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) | |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) | |
| good | 5 | $D_{4}$ | \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.i_bo_fg_nq |
| 7 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.ai_bw_ahc_wg |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_au_a_ig |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_aca_a_bwg |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ai_cm_ang_cvi |
| 23 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.i_ca_mq_ddm |
| 29 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2326 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_acq_a_dlm |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.i_bw_om_ery |
| 37 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1366 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_aca_a_cao |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.i_hc_bmm_rje |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_abs_a_gew |
| 47 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 5766 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_adw_a_inu |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_cy_a_klq |
| 59 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11286 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_afs_a_qsc |
| 61 | $D_{4}$ | \( ( 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bg_vs_keq_dohq |
| 67 | $D_{4}$ | \( ( 1 + 28 T + 328 T^{2} + 28 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.ce_cdk_bgsu_mpvu |
| 71 | $C_4\times C_2$ | \( 1 - 196 T^{2} + 18534 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_aho_a_bbkw |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_aka_a_boty |
| 79 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 12774 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_afk_a_sxi |
| 83 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 27510 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ajk_a_bosc |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 276 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.bo_bkq_vpo_jhlq |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.97.a_aoy_a_dfni |
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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
−6.93203680586600986080768054856, −6.87232700327780998496989578258, −6.48072801421408251101496233481, −6.01841265603733401399244164886, −5.91960106541200454248646014154, −5.80449359405535823943114197302, −5.66166442523399324393205179612, −5.53007184711374985178316116500, −5.04468517592485000555462171185, −4.77448940029541625188788860071, −4.74186103749307424443154411402, −4.28905763790457264791401747501, −4.18205961753310973219367231656, −4.13276606701004873527094801372, −4.11969293110976273186811240835, −3.38990137390120777782750934147, −3.11519061388371064251511633331, −3.10980225774699198379973735759, −3.09751758663445219660593593015, −1.73868977362867363481279899347, −1.59647302647266000234673884498, −1.45649566139343439212711711197, −1.45128650836431266308074633265, −0.47482921561119264598390732313, −0.33449030984361137159076865561,
0.33449030984361137159076865561, 0.47482921561119264598390732313, 1.45128650836431266308074633265, 1.45649566139343439212711711197, 1.59647302647266000234673884498, 1.73868977362867363481279899347, 3.09751758663445219660593593015, 3.10980225774699198379973735759, 3.11519061388371064251511633331, 3.38990137390120777782750934147, 4.11969293110976273186811240835, 4.13276606701004873527094801372, 4.18205961753310973219367231656, 4.28905763790457264791401747501, 4.74186103749307424443154411402, 4.77448940029541625188788860071, 5.04468517592485000555462171185, 5.53007184711374985178316116500, 5.66166442523399324393205179612, 5.80449359405535823943114197302, 5.91960106541200454248646014154, 6.01841265603733401399244164886, 6.48072801421408251101496233481, 6.87232700327780998496989578258, 6.93203680586600986080768054856
