| L(s) = 1 | − 20·25-s + 24·43-s − 2·49-s − 48·67-s − 8·73-s − 40·97-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 4·25-s + 3.65·43-s − 2/7·49-s − 5.86·67-s − 0.936·73-s − 4.06·97-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5695052350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5695052350\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.5.a_u_a_fu |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_ai_a_jy |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.13.a_u_a_qw |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.19.a_cy_a_dfi |
| 23 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ce_a_csw |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_dc_a_ewg |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ado_a_fzi |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_acy_a_gew |
| 41 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_au_a_fde |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.43.ay_oy_afwi_btva |
| 47 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_bs_a_hgo |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_aei_a_mys |
| 59 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ado_a_nle |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_agq_a_vys |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.67.bw_bro_ymy_jhuk |
| 71 | $C_2^2$ | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_jo_a_blre |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.73.i_me_cqq_cane |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_aem_a_xlm |
| 83 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_jk_a_bqkk |
| 89 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_aie_a_bobm |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) | 4.97.bo_bma_xdo_kkmw |
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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
−5.93154977825425442182171278882, −5.79614988021157417536237428513, −5.54233029288053940543261551055, −5.51604619300927957548421233171, −5.33871517378229416377493978676, −4.98241017646882015552536596833, −4.68680737016122861466814038747, −4.45350495173464985185717469403, −4.23688217135059579057438548758, −4.16657550547099855173928817144, −4.04650925150202014999146708128, −3.92342225203630921190523190495, −3.51537190599270040834843337448, −3.25789137394505289254301501619, −3.06372842229185249235608909279, −2.72936064684546984214157445525, −2.71195853719357588854875874243, −2.28621789026506313549953953071, −2.21167351928926212544674956761, −1.73203145941513375475330202270, −1.60861101164806177089251428066, −1.35887171136990757515283300413, −1.09614290018543383967700203807, −0.47297486243981105118529837094, −0.14013372588436882257199352236,
0.14013372588436882257199352236, 0.47297486243981105118529837094, 1.09614290018543383967700203807, 1.35887171136990757515283300413, 1.60861101164806177089251428066, 1.73203145941513375475330202270, 2.21167351928926212544674956761, 2.28621789026506313549953953071, 2.71195853719357588854875874243, 2.72936064684546984214157445525, 3.06372842229185249235608909279, 3.25789137394505289254301501619, 3.51537190599270040834843337448, 3.92342225203630921190523190495, 4.04650925150202014999146708128, 4.16657550547099855173928817144, 4.23688217135059579057438548758, 4.45350495173464985185717469403, 4.68680737016122861466814038747, 4.98241017646882015552536596833, 5.33871517378229416377493978676, 5.51604619300927957548421233171, 5.54233029288053940543261551055, 5.79614988021157417536237428513, 5.93154977825425442182171278882
