| L(s) = 1 | + 12·7-s + 4·9-s + 8·17-s − 16·23-s − 4·25-s + 20·31-s + 8·41-s − 20·47-s + 68·49-s + 48·63-s + 12·71-s − 16·73-s − 8·79-s − 6·81-s − 16·97-s + 24·113-s + 96·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 32·153-s + 157-s − 192·161-s + ⋯ |
| L(s) = 1 | + 4.53·7-s + 4/3·9-s + 1.94·17-s − 3.33·23-s − 4/5·25-s + 3.59·31-s + 1.24·41-s − 2.91·47-s + 68/7·49-s + 6.04·63-s + 1.42·71-s − 1.87·73-s − 0.900·79-s − 2/3·81-s − 1.62·97-s + 2.25·113-s + 8.80·119-s + 1.81·121-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 + 2.58·153-s + 0.0798·157-s − 15.1·161-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
| \(L(1)\) |
\(\approx\) |
\(5.694547348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.694547348\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_ae_a_w |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_e_a_cc |
| 7 | $D_{4}$ | \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.am_cy_amm_bmo |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_au_a_ja |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ai_cq_ang_cre |
| 19 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 1866 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_acq_a_ctu |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.23.q_hg_cai_lpu |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_aee_a_guw |
| 31 | $D_{4}$ | \( ( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.au_ki_adkm_wlu |
| 37 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abs_a_dqk |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ai_do_ayi_jbi |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_aem_a_jic |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.u_mu_evk_boja |
| 53 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_adg_a_gju |
| 59 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8618 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_afc_a_mtm |
| 61 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 7734 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_aem_a_llm |
| 67 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 25866 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_aka_a_bmgw |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.am_ky_adma_btfu |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.q_oa_fhg_ckws |
| 79 | $D_{4}$ | \( ( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.i_me_csm_cdms |
| 83 | $D_4\times C_2$ | \( 1 - 276 T^{2} + 32522 T^{4} - 276 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_akq_a_bwcw |
| 89 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ajk_a_btlu |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.q_ki_esm_cjpm |
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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
−8.048053884433717412966713175205, −8.040250212608570617062275085161, −7.83355508684672240742556567360, −7.53917559313879621791417810081, −7.38416258737571490855768464135, −6.86174036002332583599412977952, −6.83257414458794671606690733897, −6.19942722935809631447847378801, −6.01614624589092733144762643822, −5.84488552469939751394640381288, −5.55997228536087349195201798109, −5.18920524149733486870680427447, −4.98462993962920393861146330572, −4.56341897588010696309581687180, −4.41732901128861139050070445484, −4.38201926300298958803600849563, −4.26692585259710853775119678502, −3.57585239248970004918043916706, −3.34185235138908500483979879441, −2.75611734095306427474401187718, −2.22449239575256709356198648105, −1.91365119986622245651772896758, −1.71860494754989938813745621894, −1.25787348341261807773421410633, −1.10579467925375330757809358005,
1.10579467925375330757809358005, 1.25787348341261807773421410633, 1.71860494754989938813745621894, 1.91365119986622245651772896758, 2.22449239575256709356198648105, 2.75611734095306427474401187718, 3.34185235138908500483979879441, 3.57585239248970004918043916706, 4.26692585259710853775119678502, 4.38201926300298958803600849563, 4.41732901128861139050070445484, 4.56341897588010696309581687180, 4.98462993962920393861146330572, 5.18920524149733486870680427447, 5.55997228536087349195201798109, 5.84488552469939751394640381288, 6.01614624589092733144762643822, 6.19942722935809631447847378801, 6.83257414458794671606690733897, 6.86174036002332583599412977952, 7.38416258737571490855768464135, 7.53917559313879621791417810081, 7.83355508684672240742556567360, 8.040250212608570617062275085161, 8.048053884433717412966713175205
