| L(s) = 1 | − 2·4-s + 8·7-s − 12·13-s + 3·16-s + 10·25-s − 16·28-s + 16·37-s − 8·43-s + 34·49-s + 24·52-s − 4·64-s + 8·67-s + 18·73-s − 52·79-s − 96·91-s − 30·97-s − 20·100-s + 60·103-s + 16·109-s + 24·112-s − 13·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s + 3.02·7-s − 3.32·13-s + 3/4·16-s + 2·25-s − 3.02·28-s + 2.63·37-s − 1.21·43-s + 34/7·49-s + 3.32·52-s − 1/2·64-s + 0.977·67-s + 2.10·73-s − 5.85·79-s − 10.0·91-s − 3.04·97-s − 2·100-s + 5.91·103-s + 1.53·109-s + 2.26·112-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.862456791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.862456791\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 3 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ak_a_cx |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_n_a_bw |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.m_di_ro_cvb |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abi_a_bhj |
| 19 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_bm_a_bpr |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bu_a_cjb |
| 29 | $C_2^3$ | \( 1 - 23 T^{2} - 312 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_ax_a_ama |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) | 4.31.a_aeo_a_hzv |
| 37 | $C_2^2$ | \( ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.aq_eo_abnk_mdn |
| 41 | $C_2^3$ | \( 1 + 26 T^{2} - 1005 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_ba_a_abmr |
| 43 | $C_2^2$ | \( ( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.i_abm_ey_ipf |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_abc_a_gvm |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_cs_a_dcl |
| 59 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ha_a_woh |
| 61 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_fk_a_sgs |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.67.ai_lg_aclc_bsqg |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_e_a_oxy |
| 73 | $C_2^2$ | \( ( 1 - 9 T + 100 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.as_kv_aepu_bwbk |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) | 4.79.ca_bze_bfga_mwqt |
| 83 | $C_2^3$ | \( 1 - 139 T^{2} + 12432 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_afj_a_ske |
| 89 | $C_2^3$ | \( 1 - 70 T^{2} - 3021 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_acs_a_aemf |
| 97 | $C_2^2$ | \( ( 1 + 15 T + 172 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.be_vx_lyk_fgem |
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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
−7.23576182202318769637400483350, −6.95519500049058295497228695704, −6.67592312339143422220367775459, −6.40183391170559491877733701393, −5.95047781466844273960731481680, −5.77776426467531145764204439191, −5.61979285213618131718016219272, −5.24978166791980764307570772754, −5.03863332572544049951876785482, −5.00521793472112433560827518729, −4.62507169351894128553977253596, −4.61307074983977117769119259934, −4.52752512045613882204835694933, −4.11869238455494178236587778425, −4.06054966699887364616293294041, −3.43220125149618344050360414441, −3.01050181257853874251539141586, −2.99045513128380565268817874521, −2.49494748670188363720864102726, −2.37036933212909631429203548849, −1.93330307150520021880281721753, −1.77018177836370411429208267102, −1.23826749611413355540410311863, −0.896873111094220685163242369539, −0.42873801839899749518344730018,
0.42873801839899749518344730018, 0.896873111094220685163242369539, 1.23826749611413355540410311863, 1.77018177836370411429208267102, 1.93330307150520021880281721753, 2.37036933212909631429203548849, 2.49494748670188363720864102726, 2.99045513128380565268817874521, 3.01050181257853874251539141586, 3.43220125149618344050360414441, 4.06054966699887364616293294041, 4.11869238455494178236587778425, 4.52752512045613882204835694933, 4.61307074983977117769119259934, 4.62507169351894128553977253596, 5.00521793472112433560827518729, 5.03863332572544049951876785482, 5.24978166791980764307570772754, 5.61979285213618131718016219272, 5.77776426467531145764204439191, 5.95047781466844273960731481680, 6.40183391170559491877733701393, 6.67592312339143422220367775459, 6.95519500049058295497228695704, 7.23576182202318769637400483350
