| L(s) = 1 | − 4·7-s + 4·9-s + 4·19-s + 10·25-s − 28·43-s − 18·49-s + 44·61-s − 16·63-s + 20·73-s + 7·81-s + 34·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s + 16·171-s + 173-s − 40·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 4/3·9-s + 0.917·19-s + 2·25-s − 4.26·43-s − 2.57·49-s + 5.63·61-s − 2.01·63-s + 2.34·73-s + 7/9·81-s + 3.09·121-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 + 1.23·169-s + 1.22·171-s + 0.0760·173-s − 3.02·175-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.476346779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476346779\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) | |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ak_a_cx |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.7.e_bi_dk_op |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_abi_a_ul |
| 13 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_aq_a_pm |
| 17 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_acg_a_ccp |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_aca_a_cos |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.29.a_em_a_hmc |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_adk_a_fsk |
| 37 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_aei_a_iry |
| 41 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_aq_a_fbu |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) | 4.43.bc_ry_hjs_cfjn |
| 47 | $C_2^2$ | \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_ck_a_hyx |
| 53 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_bg_a_iry |
| 59 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ce_a_lly |
| 61 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) | 4.61.abs_bli_atum_hdsl |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) | 4.67.a_aeu_a_sze |
| 71 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ea_a_sxu |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) | 4.73.au_ra_ahfs_dcqd |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.79.a_ame_a_cdkg |
| 83 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_agq_a_bfik |
| 89 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_gu_a_bixe |
| 97 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_acm_a_bdje |
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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
−9.121930756973134315599662745561, −8.399486649778862251083002298731, −8.365171961237366598745488726419, −8.256413645644488743149093430888, −8.142080719348784308451884764924, −7.44375764756080496141996854807, −7.21172048854201565485589875521, −6.81673504506562909945401098139, −6.76745985099961675178430091382, −6.76158274412300164631286154622, −6.44685710099270129890573102673, −5.98239956767770817256545959499, −5.54644283656061529798686732084, −5.29443101613988514854568252647, −4.88138550035653836377424351611, −4.83087179585973810854288103502, −4.49012063105368663410121355418, −3.86332620243942542829594880227, −3.43449371691135284643896968006, −3.37197998806289309756128120454, −3.23107548140080737132588156357, −2.48930809001915653167932714819, −2.02631758669149392690401734105, −1.47855915060462285602673724242, −0.73631425203184194066012235856,
0.73631425203184194066012235856, 1.47855915060462285602673724242, 2.02631758669149392690401734105, 2.48930809001915653167932714819, 3.23107548140080737132588156357, 3.37197998806289309756128120454, 3.43449371691135284643896968006, 3.86332620243942542829594880227, 4.49012063105368663410121355418, 4.83087179585973810854288103502, 4.88138550035653836377424351611, 5.29443101613988514854568252647, 5.54644283656061529798686732084, 5.98239956767770817256545959499, 6.44685710099270129890573102673, 6.76158274412300164631286154622, 6.76745985099961675178430091382, 6.81673504506562909945401098139, 7.21172048854201565485589875521, 7.44375764756080496141996854807, 8.142080719348784308451884764924, 8.256413645644488743149093430888, 8.365171961237366598745488726419, 8.399486649778862251083002298731, 9.121930756973134315599662745561
