| L(s) = 1 | + 4·7-s + 12·13-s + 4·19-s + 2·25-s − 12·31-s + 8·37-s − 12·43-s + 10·49-s + 8·61-s − 8·67-s − 16·79-s + 48·91-s + 16·97-s + 16·103-s + 8·109-s − 4·121-s + 127-s + 131-s + 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 64·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 3.32·13-s + 0.917·19-s + 2/5·25-s − 2.15·31-s + 1.31·37-s − 1.82·43-s + 10/7·49-s + 1.02·61-s − 0.977·67-s − 1.80·79-s + 5.03·91-s + 1.62·97-s + 1.57·103-s + 0.766·109-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-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 + 4.92·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.93945061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.93945061\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 2 T^{2} + 38 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ac_a_bm |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 4 T^{2} + 38 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_e_a_bm |
| 13 | $D_{4}$ | \( ( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.am_dc_aqe_cpq |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 38 T^{2} + 822 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_bm_a_bfq |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 52 T^{2} + 1526 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_ca_a_cgs |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 86 T^{2} + 3414 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_di_a_fbi |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.31.m_fw_bpc_lde |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.37.ai_gq_abjk_ouw |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 124 T^{2} + 6998 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_eu_a_kje |
| 43 | $D_{4}$ | \( ( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.m_hs_cfs_tzy |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 158 T^{2} + 10542 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_gc_a_ppm |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 190 T^{2} + 14630 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_hi_a_vqs |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 76 T^{2} + 5078 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_cy_a_hni |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.61.ai_ki_acfo_bljy |
| 67 | $D_{4}$ | \( ( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.i_hg_bvc_bbkg |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 38 T^{2} - 3714 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_bm_a_afmw |
| 73 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_hg_a_bcvu |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.79.q_pw_fzs_dafm |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 310 T^{2} + 37790 T^{4} + 310 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_ly_a_cdxm |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 92 T^{2} + 16086 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_do_a_xus |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 158 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.aq_oq_agay_dfdm |
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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
−5.93225803142731873281283065880, −5.58658281342538487232289978129, −5.49020893736721232574616467332, −5.34303356007701459114119704414, −5.09874545954415629192598241149, −4.70601562540524376183911390628, −4.67995064444232080583014311675, −4.67589074170689324771630098510, −4.17951398311162355358130585955, −4.01767143829359694814213765314, −3.78260056896974884122769912922, −3.65786179070960783909337212316, −3.65569099067561669228976803189, −3.09944326336586371046159877865, −3.07779798130743547287580103815, −2.84610023933289797529535384858, −2.69730918150489330677421750664, −1.96667987533243909310714724049, −1.80307785238929124843503935475, −1.79845129414368275112248748901, −1.78221482958600289560253650254, −1.14812882339191650853659708606, −1.04476527953597846278367002093, −0.73908981482609976189561414505, −0.42164205873307662847978819280,
0.42164205873307662847978819280, 0.73908981482609976189561414505, 1.04476527953597846278367002093, 1.14812882339191650853659708606, 1.78221482958600289560253650254, 1.79845129414368275112248748901, 1.80307785238929124843503935475, 1.96667987533243909310714724049, 2.69730918150489330677421750664, 2.84610023933289797529535384858, 3.07779798130743547287580103815, 3.09944326336586371046159877865, 3.65569099067561669228976803189, 3.65786179070960783909337212316, 3.78260056896974884122769912922, 4.01767143829359694814213765314, 4.17951398311162355358130585955, 4.67589074170689324771630098510, 4.67995064444232080583014311675, 4.70601562540524376183911390628, 5.09874545954415629192598241149, 5.34303356007701459114119704414, 5.49020893736721232574616467332, 5.58658281342538487232289978129, 5.93225803142731873281283065880
