| L(s) = 1 | + 4·3-s + 6·9-s + 12·25-s − 4·27-s + 24·37-s − 24·47-s − 48·59-s + 48·75-s − 37·81-s − 24·83-s + 48·109-s + 96·111-s − 8·121-s + 127-s + 131-s + 137-s + 139-s − 96·141-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s − 192·177-s + 179-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s + 12/5·25-s − 0.769·27-s + 3.94·37-s − 3.50·47-s − 6.24·59-s + 5.54·75-s − 4.11·81-s − 2.63·83-s + 4.59·109-s + 9.11·111-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8.08·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-s + 0.0760·173-s − 14.4·177-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.172688694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.172688694\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| 7 | | \( 1 \) | |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.5.a_am_a_di |
| 11 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_i_a_jy |
| 13 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_bw_a_bje |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) | 4.17.a_aci_a_cew |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.19.a_acy_a_dfi |
| 23 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_ce_a_csw |
| 29 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_bc_a_cug |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_u_a_czu |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.37.ay_oa_affs_blso |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) | 4.41.a_bk_a_flu |
| 43 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_em_a_klq |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.47.y_po_ghk_bzoo |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_afs_a_qks |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.59.bw_bqi_wuq_iekc |
| 61 | $C_2^2$ | \( ( 1 + 120 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_jg_a_bgic |
| 67 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_u_a_nle |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_abo_a_pne |
| 73 | $C_2^2$ | \( ( 1 + 48 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ds_a_teo |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_abc_a_stq |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.83.y_vc_kdc_emcs |
| 89 | $C_2^2$ | \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ank_a_cqfu |
| 97 | $C_2^2$ | \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_lc_a_cgni |
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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
−6.28529455624340743276344681500, −6.26727172036099124958891431267, −5.99494124088692181499266466740, −5.86263887638049320262114602028, −5.76543016439949624170954802746, −5.11376802846146466904406921503, −5.07938130134522124117021659035, −4.65766525441782818240848838128, −4.56711300907880807425318834722, −4.55982760991144897946025918025, −4.44210978553710713871516798925, −3.87205174948274294178528533749, −3.54759432855569994749618497847, −3.53698701559598464905588566421, −3.27301797183950151530199763237, −2.96512269354220909518905118760, −2.71167593016259808640293614249, −2.70020294351996517244103150118, −2.69893896701501539368299981896, −2.06344160931186880622935419822, −1.65059741042856963922540564516, −1.55372952903544133833840960395, −1.46324327517915080267546423612, −0.71922263061413141927764647362, −0.39297994716565131233365283295,
0.39297994716565131233365283295, 0.71922263061413141927764647362, 1.46324327517915080267546423612, 1.55372952903544133833840960395, 1.65059741042856963922540564516, 2.06344160931186880622935419822, 2.69893896701501539368299981896, 2.70020294351996517244103150118, 2.71167593016259808640293614249, 2.96512269354220909518905118760, 3.27301797183950151530199763237, 3.53698701559598464905588566421, 3.54759432855569994749618497847, 3.87205174948274294178528533749, 4.44210978553710713871516798925, 4.55982760991144897946025918025, 4.56711300907880807425318834722, 4.65766525441782818240848838128, 5.07938130134522124117021659035, 5.11376802846146466904406921503, 5.76543016439949624170954802746, 5.86263887638049320262114602028, 5.99494124088692181499266466740, 6.26727172036099124958891431267, 6.28529455624340743276344681500
