| L(s) = 1 | − 5·4-s + 392·11-s − 523·16-s + 6.36e3·19-s − 7.84e3·29-s − 2.19e3·31-s − 5.55e4·41-s − 1.96e3·44-s + 4.53e4·49-s + 2.39e4·59-s − 4.87e4·61-s + 235·64-s + 1.74e5·71-s − 3.18e4·76-s − 1.30e5·79-s − 1.45e5·89-s − 1.46e5·101-s + 4.59e5·109-s + 3.92e4·116-s − 2.46e5·121-s + 1.09e4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.156·4-s + 0.976·11-s − 0.510·16-s + 4.04·19-s − 1.73·29-s − 0.409·31-s − 5.15·41-s − 0.152·44-s + 2.69·49-s + 0.894·59-s − 1.67·61-s + 0.00717·64-s + 4.11·71-s − 0.631·76-s − 2.36·79-s − 1.94·89-s − 1.43·101-s + 3.70·109-s + 0.270·116-s − 1.53·121-s + 0.0640·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.263537876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.263537876\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + 5 T^{2} + 137 p^{2} T^{4} + 5 p^{10} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 45300 T^{2} + 1039412998 T^{4} - 45300 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 196 T + 181081 T^{2} - 196 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1296980 T^{2} + 688260462198 T^{4} - 1296980 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2340610 T^{2} + 2927557675523 T^{4} - 2340610 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 3180 T + 7185073 T^{2} - 3180 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24511220 T^{2} + 233031884985798 T^{4} - 24511220 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 3920 T + 44478298 T^{2} + 3920 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 1096 T - 15343894 T^{2} + 1096 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 257567180 T^{2} + 26166130610514198 T^{4} - 257567180 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 27754 T + 414643531 T^{2} + 27754 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 37863500 T^{2} + 41125859560630998 T^{4} - 37863500 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 298326940 T^{2} + 32136931855726598 T^{4} - 298326940 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1240678540 T^{2} + 709794326287792598 T^{4} - 1240678540 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 11960 T + 1234152598 T^{2} - 11960 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 24396 T + 1596983806 T^{2} + 24396 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4294993410 T^{2} + 8014205534576787523 T^{4} - 4294993410 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 87296 T + 5453356606 T^{2} - 87296 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 5672940770 T^{2} + 16272726164284201923 T^{4} - 5672940770 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 65480 T + 5707696298 T^{2} + 65480 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 11381926610 T^{2} + 63038986097658341523 T^{4} - 11381926610 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 72810 T + 5926578523 T^{2} + 72810 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 11961289340 T^{2} + 68433641741002238598 T^{4} - 11961289340 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.84790651644835155374383816609, −7.72440938581944937335599353887, −7.61298065891296350495504165162, −6.98500188717081961901163137984, −6.93664464903077159493270715862, −6.90763930249913597638688663613, −6.64034694069303821836917608690, −5.97418006161981908271677073890, −5.68837685233168168709218745577, −5.44615516391365637469751392367, −5.27321099110046361834509340222, −5.18786885288904411912424500898, −4.69471675327669928840477821988, −4.16963355513791324110101473087, −4.01552092017270692675135101419, −3.67656684231702198026530066436, −3.18189289056459947679117846628, −3.05989958277197480007175047168, −3.01137345919572233803921218485, −1.90195304862801877968033341289, −1.89730861216422122476503051963, −1.63031512402403091216764594750, −0.997806139958791108548492616526, −0.68530553843649821238219290537, −0.36253687392018965373720377955,
0.36253687392018965373720377955, 0.68530553843649821238219290537, 0.997806139958791108548492616526, 1.63031512402403091216764594750, 1.89730861216422122476503051963, 1.90195304862801877968033341289, 3.01137345919572233803921218485, 3.05989958277197480007175047168, 3.18189289056459947679117846628, 3.67656684231702198026530066436, 4.01552092017270692675135101419, 4.16963355513791324110101473087, 4.69471675327669928840477821988, 5.18786885288904411912424500898, 5.27321099110046361834509340222, 5.44615516391365637469751392367, 5.68837685233168168709218745577, 5.97418006161981908271677073890, 6.64034694069303821836917608690, 6.90763930249913597638688663613, 6.93664464903077159493270715862, 6.98500188717081961901163137984, 7.61298065891296350495504165162, 7.72440938581944937335599353887, 7.84790651644835155374383816609
