| L(s) = 1 | + 4·9-s + 4·11-s + 4·19-s − 24·29-s + 20·31-s + 16·41-s + 4·49-s + 12·59-s − 8·61-s + 44·71-s + 40·79-s + 6·81-s + 24·89-s + 16·99-s − 8·101-s − 8·109-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 1.20·11-s + 0.917·19-s − 4.45·29-s + 3.59·31-s + 2.49·41-s + 4/7·49-s + 1.56·59-s − 1.02·61-s + 5.22·71-s + 4.50·79-s + 2/3·81-s + 2.54·89-s + 1.60·99-s − 0.796·101-s − 0.766·109-s − 2.54·121-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 − 0.153·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.821208425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.821208425\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_ae_a_k |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.7.a_ae_a_dy |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ae_bs_aeu_bcc |
| 17 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 154 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_am_a_afy |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.ae_cy_aim_dfm |
| 23 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 2810 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_adg_a_eec |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.y_lw_dym_yuo |
| 31 | $D_{4}$ | \( ( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.au_im_acsa_riw |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.37.a_afs_a_mdy |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.aq_jc_adae_xrq |
| 43 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 2730 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_ae_a_eba |
| 47 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1670 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_abk_a_cmg |
| 53 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 10934 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_aga_a_qeo |
| 59 | $D_{4}$ | \( ( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.am_fk_abzg_upe |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.i_ca_yi_oju |
| 67 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4566 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_adw_a_gtq |
| 71 | $D_{4}$ | \( ( 1 - 22 T + 260 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.abs_bmq_avoe_iipe |
| 73 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 14310 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) | 4.73.a_agi_a_vek |
| 79 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) | 4.79.abo_bii_atga_htmk |
| 83 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aem_a_zji |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ay_si_ajbk_dyda |
| 97 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 15750 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_e_a_xhu |
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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
−6.52413648097518119939058277934, −6.07613272037984285409359836736, −5.80311015249109725252798780461, −5.71036889756793595123292630285, −5.48468105688969550982404595503, −5.24638364110222647674257771930, −5.20020529548779316860454634427, −4.69990541488134432813518885055, −4.63142152048147256638405519703, −4.38036049200760685497525913072, −4.16340734505740403132813721549, −3.98682693962375277498342585113, −3.75177725679110592638333604967, −3.55173691494739512922176539917, −3.34437787594628348722915417412, −3.26355056198712329833718555201, −2.66343757657721163547170906331, −2.45595650494944013117709489156, −2.12717383347743707971496201363, −2.08325303428210443779330105910, −1.76882108668601119891515802205, −1.38411702945983372682630416221, −0.855372708101435467756346112357, −0.77684747156231216479785480763, −0.65422571237188316829540657861,
0.65422571237188316829540657861, 0.77684747156231216479785480763, 0.855372708101435467756346112357, 1.38411702945983372682630416221, 1.76882108668601119891515802205, 2.08325303428210443779330105910, 2.12717383347743707971496201363, 2.45595650494944013117709489156, 2.66343757657721163547170906331, 3.26355056198712329833718555201, 3.34437787594628348722915417412, 3.55173691494739512922176539917, 3.75177725679110592638333604967, 3.98682693962375277498342585113, 4.16340734505740403132813721549, 4.38036049200760685497525913072, 4.63142152048147256638405519703, 4.69990541488134432813518885055, 5.20020529548779316860454634427, 5.24638364110222647674257771930, 5.48468105688969550982404595503, 5.71036889756793595123292630285, 5.80311015249109725252798780461, 6.07613272037984285409359836736, 6.52413648097518119939058277934
