| L(s) = 1 | + 2·4-s − 5·16-s − 10·25-s − 20·64-s + 64·79-s − 20·100-s + 56·109-s − 44·121-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 | + 4-s − 5/4·16-s − 2·25-s − 5/2·64-s + 7.20·79-s − 2·100-s + 5.36·109-s − 4·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012518535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012518535\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| 7 | | \( 1 \) | |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_ac_a_j |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.11.a_bs_a_bby |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abc_a_bdu |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abs_a_buk |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_acq_a_dhe |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_e_a_cwc |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.37.a_afs_a_mdy |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.41.a_gi_a_oxy |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.43.a_agq_a_qks |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_abc_a_gvm |
| 53 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_gq_a_tgo |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.59.a_jc_a_bexi |
| 61 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ajc_a_bfpu |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.67.a_aki_a_bnvy |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.71.a_ky_a_bsti |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.73.a_alg_a_bvhu |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) | 4.79.acm_ctg_aburo_trja |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_lw_a_cdmc |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.89.a_ns_a_cshy |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.97.a_aoy_a_dfni |
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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
−6.34843486620318165751656061874, −6.19079399769153259307193457158, −6.11044605449425678361423783042, −6.02238580470900251719285695696, −5.62048086150329969095041601613, −5.39385584971338288181319488639, −5.04739342423436136104754712392, −5.02926206266923960015834855137, −4.77382697357999463053472783151, −4.61509444032908228072237867754, −4.19786150240645295867588234570, −4.08160129283369348909525379733, −3.77894864223420853008948776919, −3.57432714393167929764734589669, −3.44622082689234000234220375879, −3.02796370377298157288236270625, −2.92781085549024961559567063104, −2.38564979546903177981142448267, −2.14971589178965098908668060693, −2.09457329437901648461990685302, −2.09387648014195660742720991718, −1.57632567515753110429033608429, −1.03753987550979574632118561910, −0.835109260496269508668065080445, −0.16495628832501402483758919328,
0.16495628832501402483758919328, 0.835109260496269508668065080445, 1.03753987550979574632118561910, 1.57632567515753110429033608429, 2.09387648014195660742720991718, 2.09457329437901648461990685302, 2.14971589178965098908668060693, 2.38564979546903177981142448267, 2.92781085549024961559567063104, 3.02796370377298157288236270625, 3.44622082689234000234220375879, 3.57432714393167929764734589669, 3.77894864223420853008948776919, 4.08160129283369348909525379733, 4.19786150240645295867588234570, 4.61509444032908228072237867754, 4.77382697357999463053472783151, 5.02926206266923960015834855137, 5.04739342423436136104754712392, 5.39385584971338288181319488639, 5.62048086150329969095041601613, 6.02238580470900251719285695696, 6.11044605449425678361423783042, 6.19079399769153259307193457158, 6.34843486620318165751656061874
