| L(s) = 1 | + 2·4-s − 8·9-s + 3·16-s + 8·17-s − 2·25-s − 16·36-s + 16·43-s + 4·49-s − 24·53-s − 32·61-s + 12·64-s + 16·68-s + 30·81-s − 4·100-s − 8·101-s − 64·103-s − 32·107-s − 24·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s − 24·144-s + 149-s + 151-s − 64·153-s + ⋯ |
| L(s) = 1 | + 4-s − 8/3·9-s + 3/4·16-s + 1.94·17-s − 2/5·25-s − 8/3·36-s + 2.43·43-s + 4/7·49-s − 3.29·53-s − 4.09·61-s + 3/2·64-s + 1.94·68-s + 10/3·81-s − 2/5·100-s − 0.796·101-s − 6.30·103-s − 3.09·107-s − 2.25·113-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2·144-s + 0.0819·149-s + 0.0813·151-s − 5.17·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6307016586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6307016586\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| 13 | | \( 1 \) | |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) | 4.2.a_ac_a_b |
| 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_i_a_bi |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_ae_a_aba |
| 11 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_abg_a_ry |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ai_cy_aom_czu |
| 19 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 1714 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_acm_a_cny |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_dk_a_ele |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_ca_a_dms |
| 31 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 606 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_aq_a_axi |
| 37 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_ae_a_ebm |
| 41 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3654 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_acy_a_fko |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.aq_gm_acgm_rpy |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11014 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_agi_a_qhq |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.y_ky_ejo_bmdq |
| 59 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8466 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_aey_a_mnq |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.61.bg_ye_lsa_eekc |
| 67 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aka_a_bmhi |
| 71 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 10114 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_adc_a_oza |
| 73 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_afs_a_xwo |
| 79 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_gq_a_bdko |
| 83 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 27510 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ajk_a_bosc |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_aky_a_cbgw |
| 97 | $D_4\times C_2$ | \( 1 - 316 T^{2} + 43270 T^{4} - 316 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_ame_a_cmag |
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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
−7.38787876365808593939091285921, −6.98626548003212158179041785538, −6.82221162198251592563136398903, −6.68026629659112249573952724757, −6.37310602089423954763401098121, −6.04451694342105748011833703233, −5.79294103948924174409063620893, −5.74838066605880178672878990446, −5.65786232687484104842294795592, −5.29475780043167250898893301307, −5.26964277731900305591646829261, −4.56861212672698714531854720279, −4.50391389307525393612274071987, −4.31221006717434068661731417328, −3.75724676494181573123005356296, −3.50276247296023174612508716323, −3.27695256429096073333051174624, −2.91570766735454029961106518232, −2.90924621206307939805658916047, −2.57577258599422741840112032308, −2.32128960413340616336943002632, −1.77943574487807625580647405467, −1.26387986542974038084162904967, −1.21639386036139980496038552146, −0.18910554330363969961057759617,
0.18910554330363969961057759617, 1.21639386036139980496038552146, 1.26387986542974038084162904967, 1.77943574487807625580647405467, 2.32128960413340616336943002632, 2.57577258599422741840112032308, 2.90924621206307939805658916047, 2.91570766735454029961106518232, 3.27695256429096073333051174624, 3.50276247296023174612508716323, 3.75724676494181573123005356296, 4.31221006717434068661731417328, 4.50391389307525393612274071987, 4.56861212672698714531854720279, 5.26964277731900305591646829261, 5.29475780043167250898893301307, 5.65786232687484104842294795592, 5.74838066605880178672878990446, 5.79294103948924174409063620893, 6.04451694342105748011833703233, 6.37310602089423954763401098121, 6.68026629659112249573952724757, 6.82221162198251592563136398903, 6.98626548003212158179041785538, 7.38787876365808593939091285921
