| L(s) = 1 | + 6·3-s + 21·9-s − 14·13-s + 14·19-s + 54·27-s + 22·31-s + 20·37-s − 84·39-s − 10·43-s − 13·49-s + 84·57-s − 26·61-s + 26·79-s + 108·81-s + 132·93-s − 38·97-s + 40·103-s − 38·109-s + 120·111-s − 294·117-s − 44·121-s + 127-s − 60·129-s + 131-s + 137-s + 139-s − 78·147-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 7·9-s − 3.88·13-s + 3.21·19-s + 10.3·27-s + 3.95·31-s + 3.28·37-s − 13.4·39-s − 1.52·43-s − 1.85·49-s + 11.1·57-s − 3.32·61-s + 2.92·79-s + 12·81-s + 13.6·93-s − 3.85·97-s + 3.94·103-s − 3.63·109-s + 11.3·111-s − 27.1·117-s − 4·121-s + 0.0887·127-s − 5.28·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.43·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.399692133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.399692133\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) | |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.5.a_a_a_az |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) | 4.7.a_n_a_eq |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.11.a_bs_a_bby |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 22 T^{2} + p^{2} T^{4} ) \) | 4.13.o_cb_aew_abym |
| 17 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.17.a_a_a_ald |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \) | 4.19.ao_by_js_adsb |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.23.a_a_a_bos |
| 29 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.29.a_a_a_cms |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 59 T^{2} + p^{2} T^{4} ) \) | 4.31.aw_ji_acye_svf |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_ade_a_hlz |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 5 T + p T^{2} )^{2}( 1 - 61 T^{2} + p^{2} T^{4} ) \) | 4.43.k_by_agy_aeof |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.47.a_ahg_a_tpu |
| 53 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ec_a_mmd |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.59.a_a_a_afdx |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 121 T^{2} + p^{2} T^{4} ) \) | 4.61.ba_go_acia_abpcb |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) | 4.67.a_ef_a_kyi |
| 71 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_fm_a_wjr |
| 73 | $C_2^2$ | \( ( 1 + 143 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_la_a_bual |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - 142 T^{2} + p^{2} T^{4} ) \) | 4.79.aba_hd_cla_abyfw |
| 83 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_gk_a_beox |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) | 4.89.a_a_a_alsr |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 19 T + p T^{2} )^{2}( 1 + 167 T^{2} + p^{2} T^{4} ) \) | 4.97.bm_bbu_ovw_giyp |
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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
−7.83177935415582389992535979139, −7.81275116303150332363630394968, −7.68417107390136938149518762026, −7.68238330122320676024597655696, −7.43822168486282783827279753220, −6.79986938416565620677541565237, −6.69670621028378898455155362507, −6.55494915967014818488398778498, −6.40089908570384358654564358401, −5.51317404320217304683832855196, −5.49517127755256664676024835930, −4.94033603790574622855528503646, −4.89848701180947344926063676420, −4.61556138714016531312696844683, −4.29949189951232798577420508262, −4.18194976324298066619874275735, −3.64644432560875225657879954395, −3.15088367685776090174303619579, −2.95571199940987021376400353711, −2.81447436351838292959351049710, −2.68023172231866565934671332033, −2.49354749339934097325669719097, −1.81890710963619590499439279143, −1.48635667794053151497638489993, −0.887688086153105071811725621281,
0.887688086153105071811725621281, 1.48635667794053151497638489993, 1.81890710963619590499439279143, 2.49354749339934097325669719097, 2.68023172231866565934671332033, 2.81447436351838292959351049710, 2.95571199940987021376400353711, 3.15088367685776090174303619579, 3.64644432560875225657879954395, 4.18194976324298066619874275735, 4.29949189951232798577420508262, 4.61556138714016531312696844683, 4.89848701180947344926063676420, 4.94033603790574622855528503646, 5.49517127755256664676024835930, 5.51317404320217304683832855196, 6.40089908570384358654564358401, 6.55494915967014818488398778498, 6.69670621028378898455155362507, 6.79986938416565620677541565237, 7.43822168486282783827279753220, 7.68238330122320676024597655696, 7.68417107390136938149518762026, 7.81275116303150332363630394968, 7.83177935415582389992535979139
