| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s + 4·8-s − 6·9-s + 8·10-s + 8·16-s − 12·18-s − 4·19-s + 8·20-s + 24·23-s − 4·25-s + 20·29-s + 8·32-s − 12·36-s − 8·38-s + 16·40-s − 20·43-s − 24·45-s + 48·46-s + 20·49-s − 8·50-s + 8·53-s + 40·58-s + 8·64-s − 16·67-s + 48·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s + 1.41·8-s − 2·9-s + 2.52·10-s + 2·16-s − 2.82·18-s − 0.917·19-s + 1.78·20-s + 5.00·23-s − 4/5·25-s + 3.71·29-s + 1.41·32-s − 2·36-s − 1.29·38-s + 2.52·40-s − 3.04·43-s − 3.57·45-s + 7.07·46-s + 20/7·49-s − 1.13·50-s + 1.09·53-s + 5.25·58-s + 64-s − 1.95·67-s + 5.69·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.54379348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.54379348\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) | |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ae_u_aca_fy |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_au_a_hq |
| 11 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_am_a_di |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abs_a_bfq |
| 17 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1466 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_aci_a_cek |
| 19 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.e_bc_eu_bne |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.23.ay_lw_adsy_vic |
| 29 | $D_{4}$ | \( ( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.au_ie_acnk_psk |
| 31 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2970 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_acq_a_ekg |
| 41 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2790 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_acq_a_edi |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.u_me_eme_bjlq |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_gi_a_qmo |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.53.ai_jc_abye_bcss |
| 59 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aie_a_baxy |
| 61 | $D_4\times C_2$ | \( 1 + 68 T^{2} + 1686 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_cq_a_cmw |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.67.q_oa_fdo_chgo |
| 71 | $D_{4}$ | \( ( 1 - 24 T + 274 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.abw_brg_aymy_jmze |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.i_im_cbw_bipy |
| 79 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 11994 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_au_a_rti |
| 83 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 8630 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_adg_a_mty |
| 89 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 25946 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_ahw_a_bmjy |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.ai_bc_abfs_bgmo |
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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
−7.02192554855406089162372336623, −6.73248930853009747340355172178, −6.64039416734445678874498579728, −6.61019968104621794972729847005, −6.51015518331160171089781660893, −6.03253490690761630253721717637, −5.63748434339351426619926897148, −5.50550686043735181835488245219, −5.40653066857275769862985971062, −5.34247562530831811079202101263, −4.98359865847112074205689893567, −4.82093989680765355779996452224, −4.43134923400772356077887098080, −4.28245655431936700685705876208, −4.08354703267216182881588452182, −3.33696440904659457335346902161, −3.28400065902590512228719706150, −3.05086586580877887217236724393, −3.03400981939804411592480648723, −2.30820956729624527971094338171, −2.25342949529353814261914331170, −2.13947982638133250048305426329, −1.46381003402249827491135508559, −0.870997882387445428837510785556, −0.861773648917964766796216843560,
0.861773648917964766796216843560, 0.870997882387445428837510785556, 1.46381003402249827491135508559, 2.13947982638133250048305426329, 2.25342949529353814261914331170, 2.30820956729624527971094338171, 3.03400981939804411592480648723, 3.05086586580877887217236724393, 3.28400065902590512228719706150, 3.33696440904659457335346902161, 4.08354703267216182881588452182, 4.28245655431936700685705876208, 4.43134923400772356077887098080, 4.82093989680765355779996452224, 4.98359865847112074205689893567, 5.34247562530831811079202101263, 5.40653066857275769862985971062, 5.50550686043735181835488245219, 5.63748434339351426619926897148, 6.03253490690761630253721717637, 6.51015518331160171089781660893, 6.61019968104621794972729847005, 6.64039416734445678874498579728, 6.73248930853009747340355172178, 7.02192554855406089162372336623
