| L(s) = 1 | + 2·4-s − 3·5-s + 5·9-s + 14·11-s − 5·16-s − 16·19-s − 6·20-s + 5·25-s + 4·29-s − 2·31-s + 10·36-s + 4·41-s + 28·44-s − 15·45-s − 42·55-s + 20·59-s + 24·61-s − 20·64-s + 4·71-s − 32·76-s − 34·79-s + 15·80-s + 9·81-s − 20·89-s + 48·95-s + 70·99-s + 10·100-s + ⋯ |
| L(s) = 1 | + 4-s − 1.34·5-s + 5/3·9-s + 4.22·11-s − 5/4·16-s − 3.67·19-s − 1.34·20-s + 25-s + 0.742·29-s − 0.359·31-s + 5/3·36-s + 0.624·41-s + 4.22·44-s − 2.23·45-s − 5.66·55-s + 2.60·59-s + 3.07·61-s − 5/2·64-s + 0.474·71-s − 3.67·76-s − 3.82·79-s + 1.67·80-s + 81-s − 2.11·89-s + 4.92·95-s + 7.03·99-s + 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.633320984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.633320984\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) | |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_ac_a_j |
| 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) | 4.3.a_af_a_q |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) | 4.7.a_a_a_abi |
| 11 | $D_{4}$ | \( ( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ao_dx_aty_cyu |
| 13 | $D_4\times C_2$ | \( 1 - 33 T^{2} + 536 T^{4} - 33 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_abh_a_uq |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_abo_a_bgo |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.19.q_gq_bsy_izm |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_acq_a_dhe |
| 31 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.c_ef_go_hgm |
| 37 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 950 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abk_a_bko |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ae_ea_aoa_jec |
| 43 | $D_4\times C_2$ | \( 1 - 21 T^{2} + 1952 T^{4} - 21 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_av_a_cxc |
| 47 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 2904 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abl_a_ehs |
| 53 | $D_4\times C_2$ | \( 1 - 193 T^{2} + 14856 T^{4} - 193 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ahl_a_vzk |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 110 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.au_mi_afaa_btra |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.61.ay_rs_ahue_cvyc |
| 67 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 17006 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ahk_a_zec |
| 71 | $D_{4}$ | \( ( 1 - 2 T + 110 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.ae_iq_abbw_bhra |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_aho_a_bdzi |
| 79 | $D_{4}$ | \( ( 1 + 17 T + 222 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.bi_bcf_pdq_gcxw |
| 83 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 5814 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_bs_a_ipq |
| 89 | $D_{4}$ | \( ( 1 + 10 T + 170 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.u_qy_hrg_donm |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_alg_a_chjq |
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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
−9.558636311981072011114231636562, −9.380458379422572763597515194258, −9.095691124682337976065135095210, −8.622848085982802991382927277386, −8.493721109413056414324877490466, −8.421934579477768410399368281305, −8.191944651778505599496357084967, −7.28769397468050290634388321164, −7.11927841538927821800191863085, −7.10997106416035410010324836321, −6.69998092458973117941495804480, −6.68312965066904745997099747109, −6.47571411923543533438776391264, −6.00791994071991985641030865922, −5.74470594266988277578615724227, −4.92691576994058209114940550641, −4.41627532927648895385927444905, −4.23628107313601551838872069825, −4.14871669065862588100463645526, −3.91669487215032119232006392549, −3.66443878976027824227787146903, −2.71425187992633401814744734436, −2.23372945886443880309400720295, −1.73127387610195797970364877272, −1.22484106305275691711494156090,
1.22484106305275691711494156090, 1.73127387610195797970364877272, 2.23372945886443880309400720295, 2.71425187992633401814744734436, 3.66443878976027824227787146903, 3.91669487215032119232006392549, 4.14871669065862588100463645526, 4.23628107313601551838872069825, 4.41627532927648895385927444905, 4.92691576994058209114940550641, 5.74470594266988277578615724227, 6.00791994071991985641030865922, 6.47571411923543533438776391264, 6.68312965066904745997099747109, 6.69998092458973117941495804480, 7.10997106416035410010324836321, 7.11927841538927821800191863085, 7.28769397468050290634388321164, 8.191944651778505599496357084967, 8.421934579477768410399368281305, 8.493721109413056414324877490466, 8.622848085982802991382927277386, 9.095691124682337976065135095210, 9.380458379422572763597515194258, 9.558636311981072011114231636562
