| L(s) = 1 | − 2·4-s + 9·9-s − 18·11-s + 3·16-s + 14·19-s + 10·25-s + 12·29-s − 22·31-s − 18·36-s − 18·41-s + 36·44-s + 49-s + 24·59-s − 2·61-s − 4·64-s + 6·71-s − 28·76-s + 4·79-s + 44·81-s + 12·89-s − 162·99-s − 20·100-s − 24·101-s + 10·109-s − 24·116-s + 161·121-s + 44·124-s + ⋯ |
| L(s) = 1 | − 4-s + 3·9-s − 5.42·11-s + 3/4·16-s + 3.21·19-s + 2·25-s + 2.22·29-s − 3.95·31-s − 3·36-s − 2.81·41-s + 5.42·44-s + 1/7·49-s + 3.12·59-s − 0.256·61-s − 1/2·64-s + 0.712·71-s − 3.21·76-s + 0.450·79-s + 44/9·81-s + 1.27·89-s − 16.2·99-s − 2·100-s − 2.38·101-s + 0.957·109-s − 2.22·116-s + 14.6·121-s + 3.95·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.077250167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.077250167\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 37 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) | 4.3.a_aj_a_bl |
| 7 | $D_4\times C_2$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_ab_a_ad |
| 11 | $C_4$ | \( ( 1 + 9 T + 41 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.s_gh_bka_fmn |
| 13 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 833 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_abt_a_bgb |
| 17 | $D_4\times C_2$ | \( 1 - 33 T^{2} + 569 T^{4} - 33 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_abh_a_vx |
| 19 | $C_4$ | \( ( 1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ao_ex_abfg_gbx |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.am_dc_axo_ghq |
| 31 | $C_4$ | \( ( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.w_kx_dqu_xqv |
| 37 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 1518 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_abo_a_cgk |
| 41 | $D_{4}$ | \( ( 1 + 9 T + 101 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.s_kx_dui_bdxd |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) | 4.43.a_a_a_dby |
| 47 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 494 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abw_a_ta |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ahw_a_xsg |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.59.ay_rk_ahoq_csnm |
| 61 | $D_{4}$ | \( ( 1 + T + 111 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.c_ip_ng_bdkv |
| 67 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 23198 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ajg_a_biig |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 143 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.ag_lj_abxk_bvbl |
| 73 | $D_4\times C_2$ | \( 1 + 36 T^{2} - 538 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_bk_a_aus |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.ae_iy_abds_bmqg |
| 83 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 27998 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ajg_a_bpkw |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.am_mi_aecq_ckti |
| 97 | $D_4\times C_2$ | \( 1 + 39 T^{2} + 17297 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_bn_a_zph |
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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
−8.739100997259001296618798265342, −8.617506485191786541322690274082, −8.486014673715489236286559024824, −7.86321153435029193482324787266, −7.76983747648852652437570480047, −7.76480017470726334006662736190, −7.32451563933513431550051456815, −7.21901164642356176836931676655, −6.99715725445727066908852194873, −6.69561740979144057830978451218, −6.22090002616129268893278144765, −5.42614038066760378104228618650, −5.27036176480170043043237402166, −5.20972006853873221168916111520, −5.16174315681978554591801592176, −5.02263458193672447892957030321, −4.52986418414509946267787643482, −3.97501786234077577623805799731, −3.72591792737666733580355794781, −3.09290535581912478840257014229, −3.09085984968487180357932001720, −2.58276374650143322805870526580, −2.03867292215223658109292997604, −1.41485660603361830762438369361, −0.63326505294999348003538572031,
0.63326505294999348003538572031, 1.41485660603361830762438369361, 2.03867292215223658109292997604, 2.58276374650143322805870526580, 3.09085984968487180357932001720, 3.09290535581912478840257014229, 3.72591792737666733580355794781, 3.97501786234077577623805799731, 4.52986418414509946267787643482, 5.02263458193672447892957030321, 5.16174315681978554591801592176, 5.20972006853873221168916111520, 5.27036176480170043043237402166, 5.42614038066760378104228618650, 6.22090002616129268893278144765, 6.69561740979144057830978451218, 6.99715725445727066908852194873, 7.21901164642356176836931676655, 7.32451563933513431550051456815, 7.76480017470726334006662736190, 7.76983747648852652437570480047, 7.86321153435029193482324787266, 8.486014673715489236286559024824, 8.617506485191786541322690274082, 8.739100997259001296618798265342
