| L(s) = 1 | + 3·3-s − 6·11-s − 6·17-s − 6·19-s − 3·23-s − 12·27-s + 12·29-s + 3·31-s − 18·33-s + 6·37-s − 3·41-s + 9·43-s + 18·47-s − 18·51-s − 6·53-s − 18·57-s + 3·59-s − 9·61-s − 12·67-s − 9·69-s + 9·71-s − 12·73-s − 3·79-s − 18·81-s + 36·87-s − 6·89-s + 9·93-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.80·11-s − 1.45·17-s − 1.37·19-s − 0.625·23-s − 2.30·27-s + 2.22·29-s + 0.538·31-s − 3.13·33-s + 0.986·37-s − 0.468·41-s + 1.37·43-s + 2.62·47-s − 2.52·51-s − 0.824·53-s − 2.38·57-s + 0.390·59-s − 1.15·61-s − 1.46·67-s − 1.08·69-s + 1.06·71-s − 1.40·73-s − 0.337·79-s − 2·81-s + 3.85·87-s − 0.635·89-s + 0.933·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 5 p T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.3.ad_j_ap |
| 11 | $A_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 113 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.g_bk_ej |
| 13 | $C_6$ | \( 1 + 19 T^{3} + p^{3} T^{6} \) | 3.13.a_a_t |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 36 T^{2} + 185 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.g_bk_hd |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 48 T^{2} + 211 T^{3} + 48 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.g_bw_id |
| 23 | $A_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 155 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.d_bz_fz |
| 29 | $A_4\times C_2$ | \( 1 - 12 T + 126 T^{2} - 715 T^{3} + 126 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.am_ew_abbn |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 57 T^{2} - 167 T^{3} + 57 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ad_cf_agl |
| 37 | $A_4\times C_2$ | \( 1 - 6 T + 84 T^{2} - 393 T^{3} + 84 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_dg_apd |
| 41 | $A_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 189 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.d_eb_hh |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 117 T^{2} - 595 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aj_en_awx |
| 47 | $A_4\times C_2$ | \( 1 - 18 T + 228 T^{2} - 1799 T^{3} + 228 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.as_iu_acrf |
| 53 | $A_4\times C_2$ | \( 1 + 6 T - 12 T^{2} - 441 T^{3} - 12 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_am_aqz |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 153 T^{2} - 301 T^{3} + 153 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ad_fx_alp |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 1135 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.j_en_brr |
| 67 | $A_4\times C_2$ | \( 1 + 12 T + 141 T^{2} + 1024 T^{3} + 141 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.m_fl_bnk |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 132 T^{2} - 765 T^{3} + 132 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aj_fc_abdl |
| 73 | $A_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 1560 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.m_il_cia |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 471 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_jd_sd |
| 83 | $A_4\times C_2$ | \( 1 + 192 T^{2} - 163 T^{3} + 192 p T^{4} + p^{3} T^{6} \) | 3.83.a_hk_agh |
| 89 | $A_4\times C_2$ | \( 1 + 6 T + 150 T^{2} + 369 T^{3} + 150 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_fu_of |
| 97 | $A_4\times C_2$ | \( 1 + 21 T + 357 T^{2} + 3607 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.v_nt_fit |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.40234304660443011135303142321, −6.77493235510581690083978697855, −6.65267516672994413590572489853, −6.49843196725302169508115904611, −6.21267182317372865433594322264, −5.94992670156356833382814111034, −5.88832964054461297846801020641, −5.42969454776618584537556489048, −5.24590872597739612325056216555, −5.11831958867632950932164178219, −4.62387025791387988435662574300, −4.49497249112013028216344757689, −4.34325941810491594285374216766, −3.90536130971704486701089489212, −3.76219029723440544739183061946, −3.74917766415344348367658205910, −2.89241692564788923364599698721, −2.83475853055525706658159245033, −2.74031035322100761770630360057, −2.52790453581539812220270093040, −2.36572517393913797427021435071, −2.33267193880930152768464178574, −1.56132436576316862353690537412, −1.31967213116455425342763077113, −1.02396406896554989197772382718, 0, 0, 0,
1.02396406896554989197772382718, 1.31967213116455425342763077113, 1.56132436576316862353690537412, 2.33267193880930152768464178574, 2.36572517393913797427021435071, 2.52790453581539812220270093040, 2.74031035322100761770630360057, 2.83475853055525706658159245033, 2.89241692564788923364599698721, 3.74917766415344348367658205910, 3.76219029723440544739183061946, 3.90536130971704486701089489212, 4.34325941810491594285374216766, 4.49497249112013028216344757689, 4.62387025791387988435662574300, 5.11831958867632950932164178219, 5.24590872597739612325056216555, 5.42969454776618584537556489048, 5.88832964054461297846801020641, 5.94992670156356833382814111034, 6.21267182317372865433594322264, 6.49843196725302169508115904611, 6.65267516672994413590572489853, 6.77493235510581690083978697855, 7.40234304660443011135303142321
