| L(s) = 1 | + 2·3-s + 6·5-s − 4·7-s + 4·9-s + 12·11-s − 12·13-s + 12·15-s + 6·17-s + 6·19-s − 8·21-s + 11·25-s + 4·27-s + 24·33-s − 24·35-s − 2·37-s − 24·39-s − 6·41-s + 24·45-s − 12·47-s + 18·49-s + 12·51-s − 12·53-s + 72·55-s + 12·57-s + 12·59-s + 6·61-s − 16·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.68·5-s − 1.51·7-s + 4/3·9-s + 3.61·11-s − 3.32·13-s + 3.09·15-s + 1.45·17-s + 1.37·19-s − 1.74·21-s + 11/5·25-s + 0.769·27-s + 4.17·33-s − 4.05·35-s − 0.328·37-s − 3.84·39-s − 0.937·41-s + 3.57·45-s − 1.75·47-s + 18/7·49-s + 1.68·51-s − 1.64·53-s + 9.70·55-s + 1.58·57-s + 1.56·59-s + 0.768·61-s − 2.01·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.160710228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.160710228\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 4 T^{3} - 5 T^{4} + 4 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.3.ac_a_e_af |
| 5 | $C_2^2$ | \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ag_z_ada_hw |
| 7 | $C_2^2$ | \( ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.e_ac_q_gh |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.am_do_asa_cry |
| 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 | $D_4\times C_2$ | \( 1 - 6 T + 13 T^{2} - 6 T^{3} - 84 T^{4} - 6 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ag_n_ag_adg |
| 19 | $D_4\times C_2$ | \( 1 - 6 T + 44 T^{2} - 192 T^{3} + 891 T^{4} - 192 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ag_bs_ahk_bih |
| 23 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2106 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_acq_a_dda |
| 29 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_bi_a_cxv |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4314 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_adw_a_gjy |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 7 T^{2} - 234 T^{3} - 1308 T^{4} - 234 p T^{5} - 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.g_ah_aja_abyi |
| 43 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 3210 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_ae_a_aetm |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.m_jc_cpw_bais |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 14 T^{2} + 288 T^{3} + 6459 T^{4} + 288 p T^{5} + 14 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.m_o_lc_jol |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 142 T^{2} - 1128 T^{3} + 8187 T^{4} - 1128 p T^{5} + 142 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.am_fm_abrk_mcx |
| 61 | $C_2^2$ | \( ( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.ag_fh_abcw_ssa |
| 67 | $D_4\times C_2$ | \( 1 + 10 T - 32 T^{2} - 20 T^{3} + 6235 T^{4} - 20 p T^{5} - 32 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.k_abg_au_jfv |
| 71 | $C_2^3$ | \( 1 - 130 T^{2} + 11859 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_afa_a_rod |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.73.aq_oy_afoq_cqks |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 52 T^{2} - 384 T^{3} - 1197 T^{4} - 384 p T^{5} - 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.g_aca_aou_abub |
| 83 | $D_4\times C_2$ | \( 1 - 6 T - 64 T^{2} + 396 T^{3} + 123 T^{4} + 396 p T^{5} - 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.ag_acm_pg_et |
| 89 | $D_4\times C_2$ | \( 1 - 18 T + 169 T^{2} - 1098 T^{3} + 5412 T^{4} - 1098 p T^{5} + 169 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.as_gn_abqg_iae |
| 97 | $D_4\times C_2$ | \( 1 - 310 T^{2} + 42411 T^{4} - 310 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_aly_a_cktf |
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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
−7.47664810114519235063165715360, −7.41458148162080071685790421824, −7.30879155180185177164415339659, −6.89553211761516122233479477996, −6.86524692381338767908854037187, −6.35849650452927331400811528648, −6.26464824539664817254652807506, −6.16453483572719737902115581695, −6.02081712396043792849707111558, −5.46724427569349270591732648391, −5.18494800891820305635030744786, −5.11565785556131063390584949370, −4.84800040595270504878455919636, −4.48262653937140746380588480815, −3.97039160964177468677306455853, −3.82697658892244715894058188395, −3.48851058270205280070355394743, −3.28875383947870717323617446545, −3.05548064344794191954266882416, −2.52637846742919298994091458937, −2.27047471106595560932750914254, −1.91847719082723010153141443396, −1.75892190641303370850394615058, −1.31319466683809854682714035615, −0.78425420101074296600728772175,
0.78425420101074296600728772175, 1.31319466683809854682714035615, 1.75892190641303370850394615058, 1.91847719082723010153141443396, 2.27047471106595560932750914254, 2.52637846742919298994091458937, 3.05548064344794191954266882416, 3.28875383947870717323617446545, 3.48851058270205280070355394743, 3.82697658892244715894058188395, 3.97039160964177468677306455853, 4.48262653937140746380588480815, 4.84800040595270504878455919636, 5.11565785556131063390584949370, 5.18494800891820305635030744786, 5.46724427569349270591732648391, 6.02081712396043792849707111558, 6.16453483572719737902115581695, 6.26464824539664817254652807506, 6.35849650452927331400811528648, 6.86524692381338767908854037187, 6.89553211761516122233479477996, 7.30879155180185177164415339659, 7.41458148162080071685790421824, 7.47664810114519235063165715360
