| L(s) = 1 | + 4·2-s + 10·4-s + 4·5-s − 4·7-s + 20·8-s − 6·9-s + 16·10-s + 8·11-s − 4·13-s − 16·14-s + 35·16-s + 4·17-s − 24·18-s + 8·19-s + 40·20-s + 32·22-s − 2·25-s − 16·26-s + 4·27-s − 40·28-s − 4·29-s + 56·32-s + 16·34-s − 16·35-s − 60·36-s − 8·37-s + 32·38-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 1.78·5-s − 1.51·7-s + 7.07·8-s − 2·9-s + 5.05·10-s + 2.41·11-s − 1.10·13-s − 4.27·14-s + 35/4·16-s + 0.970·17-s − 5.65·18-s + 1.83·19-s + 8.94·20-s + 6.82·22-s − 2/5·25-s − 3.13·26-s + 0.769·27-s − 7.55·28-s − 0.742·29-s + 9.89·32-s + 2.74·34-s − 2.70·35-s − 10·36-s − 1.31·37-s + 5.19·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(98.63409367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(98.63409367\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 23 | | \( 1 \) | |
| good | 3 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 p T^{2} - 4 T^{3} + 20 T^{4} - 4 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \) | 4.3.a_g_ae_u |
| 5 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 18 T^{2} - 8 p T^{3} + 116 T^{4} - 8 p^{2} T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ae_s_abo_em |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ai_cm_aku_bta |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 40 T^{2} + 12 p T^{3} + 710 T^{4} + 12 p^{2} T^{5} + 40 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.e_bo_ga_bbi |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 66 T^{2} - 200 T^{3} + 1668 T^{4} - 200 p T^{5} + 66 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ae_co_ahs_cme |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 4 p T^{2} - 424 T^{3} + 2182 T^{4} - 424 p T^{5} + 4 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ai_cy_aqi_dfy |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 60 T^{2} - 28 T^{3} + 1194 T^{4} - 28 p T^{5} + 60 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.e_ci_abc_bty |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 86 T^{2} - 12 T^{3} + 3508 T^{4} - 12 p T^{5} + 86 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_di_am_fey |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 128 T^{2} + 808 T^{3} + 6898 T^{4} + 808 p T^{5} + 128 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.i_ey_bfc_kfi |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 68 T^{2} + 256 T^{3} + 2102 T^{4} + 256 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_cq_jw_dcw |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 68 T^{2} - 680 T^{3} + 4998 T^{4} - 680 p T^{5} + 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ai_cq_abae_hkg |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 78 T^{2} + 532 T^{3} + 2228 T^{4} + 532 p T^{5} + 78 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_da_um_dhs |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 88 T^{2} + 7042 T^{4} + 88 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_dk_a_kkw |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 222 T^{2} - 1284 T^{3} + 19172 T^{4} - 1284 p T^{5} + 222 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.ai_io_abxk_bcjk |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 186 T^{2} - 1488 T^{3} + 14484 T^{4} - 1488 p T^{5} + 186 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.am_he_acfg_vlc |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 32 T + 608 T^{2} - 7712 T^{3} + 73458 T^{4} - 7712 p T^{5} + 608 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.abg_xk_alkq_eeri |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 244 T^{2} - 1672 T^{3} + 24598 T^{4} - 1672 p T^{5} + 244 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ai_jk_acmi_bkkc |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 72 T^{2} - 524 T^{3} + 5622 T^{4} - 524 p T^{5} + 72 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ae_cu_aue_iig |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 128 T^{2} - 1008 T^{3} + 9570 T^{4} - 1008 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.aq_ey_abmu_oec |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T + 464 T^{2} + 5332 T^{3} + 63574 T^{4} + 5332 p T^{5} + 464 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.u_rw_hxc_dqbe |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 28 T + 362 T^{2} - 2224 T^{3} + 12340 T^{4} - 2224 p T^{5} + 362 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.abc_ny_adho_sgq |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 374 T^{2} + 4564 T^{3} + 53676 T^{4} + 4564 p T^{5} + 374 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.q_ok_gto_dbkm |
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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
−5.57580813739842049822075090422, −5.28010713135878994129075359876, −5.25775588678802209305415321760, −5.16182490403579519694613486305, −5.10984088735742219999490265083, −4.51969501418063769287504193870, −4.42213948259137994993576666544, −4.27416015875402938522839564619, −3.89443902838383760838681876282, −3.64918563061939980492914108400, −3.62634617984024371005067912803, −3.59586152037021013648216273588, −3.45987467291543737831104210185, −2.98054158376695139819435748026, −2.87279505135269703625374408726, −2.73893119531073135994809070274, −2.64625881869803142309015420814, −2.10611567259679401947539429418, −2.05084942454454073425382699537, −1.91281510791106754367079275469, −1.81576005400969385547414205920, −1.33171725538161297199966724012, −0.77238242667347344334995423097, −0.77069789455870825583072016594, −0.57454605904393982169827302657,
0.57454605904393982169827302657, 0.77069789455870825583072016594, 0.77238242667347344334995423097, 1.33171725538161297199966724012, 1.81576005400969385547414205920, 1.91281510791106754367079275469, 2.05084942454454073425382699537, 2.10611567259679401947539429418, 2.64625881869803142309015420814, 2.73893119531073135994809070274, 2.87279505135269703625374408726, 2.98054158376695139819435748026, 3.45987467291543737831104210185, 3.59586152037021013648216273588, 3.62634617984024371005067912803, 3.64918563061939980492914108400, 3.89443902838383760838681876282, 4.27416015875402938522839564619, 4.42213948259137994993576666544, 4.51969501418063769287504193870, 5.10984088735742219999490265083, 5.16182490403579519694613486305, 5.25775588678802209305415321760, 5.28010713135878994129075359876, 5.57580813739842049822075090422
