| L(s) = 1 | + 10·9-s − 2·25-s − 28·49-s + 57·81-s − 36·89-s − 68·97-s + 84·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 10/3·9-s − 2/5·25-s − 4·49-s + 19/3·81-s − 3.81·89-s − 6.90·97-s + 7.90·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.022235943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.022235943\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_ak_a_br |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \) | 4.5.a_c_a_bz |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.13.a_ca_a_bna |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.19.a_acy_a_dfi |
| 23 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_cs_a_djv |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_acw_a_ewp |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) | 4.37.a_by_a_ezj |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.43.a_agq_a_qks |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_dw_a_kgc |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.53.a_fk_a_poo |
| 59 | $C_2^2$ | \( ( 1 + 107 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ig_a_bbgd |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.61.a_jk_a_bhas |
| 67 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_cs_a_pcl |
| 71 | $C_2^2$ | \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_akg_a_bpcd |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.73.a_alg_a_bvhu |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.83.a_amu_a_cjdu |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) | 4.89.bk_bgk_snw_hzzn |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{4} \) | 4.97.cq_ddq_cgiy_bbcrz |
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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
−7.41632123767573155421672669254, −7.20945507078120452430924827665, −7.08717308095085055506520465989, −6.91278791034619673213446275734, −6.62779944096977045105103078099, −6.35914927337639734128076888092, −6.35553066278514386306827235256, −5.64444357619983957780933515194, −5.64273758030996462760064359307, −5.57011545533422889164640676578, −4.82591417845408163399144053977, −4.71863549584161949735944126655, −4.60158185228490121108705080507, −4.47243686332312647600881720756, −4.14294938527204085750894915083, −3.72334471585182848877686772002, −3.56472862505509108100027456113, −3.37779323210341222235928492768, −2.86606996871579621046284981804, −2.54341735341400171958358144988, −2.13613264212526906966500590315, −1.52869726139392112047103671197, −1.50342371573976204007966556162, −1.39898128828889417030763336766, −0.46824775624231513194824162986,
0.46824775624231513194824162986, 1.39898128828889417030763336766, 1.50342371573976204007966556162, 1.52869726139392112047103671197, 2.13613264212526906966500590315, 2.54341735341400171958358144988, 2.86606996871579621046284981804, 3.37779323210341222235928492768, 3.56472862505509108100027456113, 3.72334471585182848877686772002, 4.14294938527204085750894915083, 4.47243686332312647600881720756, 4.60158185228490121108705080507, 4.71863549584161949735944126655, 4.82591417845408163399144053977, 5.57011545533422889164640676578, 5.64273758030996462760064359307, 5.64444357619983957780933515194, 6.35553066278514386306827235256, 6.35914927337639734128076888092, 6.62779944096977045105103078099, 6.91278791034619673213446275734, 7.08717308095085055506520465989, 7.20945507078120452430924827665, 7.41632123767573155421672669254
