| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 3·9-s + 18·13-s + 35·16-s + 12·18-s − 2·19-s − 2·25-s + 72·26-s + 56·32-s + 30·36-s − 8·38-s + 8·43-s − 10·47-s − 2·49-s − 8·50-s + 180·52-s − 18·53-s − 18·59-s + 84·64-s + 12·67-s + 60·72-s − 20·76-s − 7·81-s − 16·83-s + 32·86-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s + 9-s + 4.99·13-s + 35/4·16-s + 2.82·18-s − 0.458·19-s − 2/5·25-s + 14.1·26-s + 9.89·32-s + 5·36-s − 1.29·38-s + 1.21·43-s − 1.45·47-s − 2/7·49-s − 1.13·50-s + 24.9·52-s − 2.47·53-s − 2.34·59-s + 21/2·64-s + 1.46·67-s + 7.07·72-s − 2.29·76-s − 7/9·81-s − 1.75·83-s + 3.45·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(50.18975305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(50.18975305\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_ad_a_q |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_abk_a_vu |
| 13 | $D_{4}$ | \( ( 1 - 9 T + 42 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.as_gj_abmc_gfw |
| 19 | $D_{4}$ | \( ( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.c_cr_ec_cvs |
| 23 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_abo_a_bgo |
| 29 | $D_4\times C_2$ | \( 1 + 37 T^{2} + 796 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_bl_a_beq |
| 31 | $D_4\times C_2$ | \( 1 - 115 T^{2} + 5224 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_ael_a_hsy |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.37.a_afk_a_lhu |
| 41 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 286 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_abg_a_la |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ai_ci_aua_ifu |
| 47 | $D_{4}$ | \( ( 1 + 5 T + 96 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.k_ij_cda_xqu |
| 53 | $D_{4}$ | \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.s_jx_dtq_bgme |
| 59 | $D_{4}$ | \( ( 1 + 9 T + 134 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.s_nl_fdq_bzaa |
| 61 | $D_4\times C_2$ | \( 1 - 167 T^{2} + 14376 T^{4} - 167 p^{2} T^{6} + p^{4} T^{8} \) | 4.61.a_agl_a_vgy |
| 67 | $D_{4}$ | \( ( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.am_lc_adlc_brxm |
| 71 | $D_4\times C_2$ | \( 1 - 263 T^{2} + 27268 T^{4} - 263 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_akd_a_boiu |
| 73 | $D_4\times C_2$ | \( 1 - 79 T^{2} + 12112 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_adb_a_rxw |
| 79 | $D_4\times C_2$ | \( 1 + 40 T^{2} - 2418 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_bo_a_adpa |
| 83 | $D_{4}$ | \( ( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.q_lg_erg_cdik |
| 89 | $D_{4}$ | \( ( 1 - 29 T + 384 T^{2} - 29 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.acg_cjx_abope_rvai |
| 97 | $D_4\times C_2$ | \( 1 - 287 T^{2} + 37536 T^{4} - 287 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_alb_a_cdns |
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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
−6.80038643661279316039393411856, −6.42174824727889133241535113633, −6.29162296578601604915594667705, −6.28833068565171375306092832741, −6.24286556845818955563485487175, −5.74408023074737815990872906790, −5.71556044284375399472767245475, −5.55031059914493872132135177392, −5.14877067464327133446570787917, −4.73327024028700575291526488980, −4.59557479469362459148177043494, −4.49589534422908332852091281908, −4.26117357562893845102076822830, −3.96581623548054619202111947468, −3.75545479470995677164604421039, −3.34642901637349961705305251909, −3.33607653824191587358184691724, −3.22208925960849048035234144940, −3.10503027241112522513237359661, −2.24276695435536326466209891274, −2.08025894777975806749967251769, −1.72498835150788332891551979540, −1.61110676305463983165510145897, −0.993738097288097621508231177617, −0.989941302245934697889475420232,
0.989941302245934697889475420232, 0.993738097288097621508231177617, 1.61110676305463983165510145897, 1.72498835150788332891551979540, 2.08025894777975806749967251769, 2.24276695435536326466209891274, 3.10503027241112522513237359661, 3.22208925960849048035234144940, 3.33607653824191587358184691724, 3.34642901637349961705305251909, 3.75545479470995677164604421039, 3.96581623548054619202111947468, 4.26117357562893845102076822830, 4.49589534422908332852091281908, 4.59557479469362459148177043494, 4.73327024028700575291526488980, 5.14877067464327133446570787917, 5.55031059914493872132135177392, 5.71556044284375399472767245475, 5.74408023074737815990872906790, 6.24286556845818955563485487175, 6.28833068565171375306092832741, 6.29162296578601604915594667705, 6.42174824727889133241535113633, 6.80038643661279316039393411856
