| L(s) = 1 | + 4·4-s − 12·11-s + 9·16-s + 8·19-s − 12·29-s + 4·31-s − 24·41-s − 48·44-s + 16·49-s − 24·59-s − 8·61-s + 12·64-s + 72·71-s + 32·76-s − 16·79-s + 9·81-s + 24·89-s − 24·101-s + 32·109-s − 48·116-s + 92·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 2·4-s − 3.61·11-s + 9/4·16-s + 1.83·19-s − 2.22·29-s + 0.718·31-s − 3.74·41-s − 7.23·44-s + 16/7·49-s − 3.12·59-s − 1.02·61-s + 3/2·64-s + 8.54·71-s + 3.67·76-s − 1.80·79-s + 81-s + 2.54·89-s − 2.38·101-s + 3.06·109-s − 4.45·116-s + 8.36·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.370170491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.370170491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T^{2} + 7 T^{4} - p^{2} T^{6} + T^{8} - p^{4} T^{10} + 7 p^{4} T^{12} - p^{8} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 16 T^{2} + 121 T^{4} - 592 T^{6} + 3280 T^{8} - 592 p^{2} T^{10} + 121 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 20 T^{2} + 231 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 40 T^{2} + 786 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 - 88 T^{2} + 4753 T^{4} - 170104 T^{6} + 4569664 T^{8} - 170104 p^{2} T^{10} + 4753 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 6 T - 19 T^{2} - 18 T^{3} + 1140 T^{4} - 18 p T^{5} - 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T - 56 T^{2} + 4 T^{3} + 2515 T^{4} + 4 p T^{5} - 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 12 T + 29 T^{2} + 396 T^{3} + 5640 T^{4} + 396 p T^{5} + 29 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 88 T^{2} + 3838 T^{4} - 18304 T^{6} - 2480621 T^{8} - 18304 p^{2} T^{10} + 3838 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 160 T^{2} + 14809 T^{4} - 1019680 T^{6} + 55015600 T^{8} - 1019680 p^{2} T^{10} + 14809 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 196 T^{2} + 15174 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 12 T + 2 T^{2} + 288 T^{3} + 8187 T^{4} + 288 p T^{5} + 2 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T - 35 T^{2} - 284 T^{3} - 2096 T^{4} - 284 p T^{5} - 35 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 184 T^{2} + 16657 T^{4} - 1512664 T^{6} + 123674896 T^{8} - 1512664 p^{2} T^{10} + 16657 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 8 T - 98 T^{2} + 32 T^{3} + 14947 T^{4} + 32 p T^{5} - 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 160 T^{2} + 5929 T^{4} - 11360 p T^{6} + 23680 p^{2} T^{8} - 11360 p^{3} T^{10} + 5929 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 52 T^{2} - 12902 T^{4} + 167024 T^{6} + 129576019 T^{8} + 167024 p^{2} T^{10} - 12902 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.50942299647231440766772225672, −5.45112930238953392854414375085, −5.35697285724888169395945063147, −5.14365787047990759662960074044, −4.99231730926276296224480984720, −4.90838852424776084526549399267, −4.72084020697547006275754002554, −4.64730575345348049159149528707, −4.34512017051328941468936154739, −4.09678938218898410482516246316, −3.66816164342689654726684376235, −3.61742932060607815507210780877, −3.57776446103860760857071855834, −3.37643789070114064119387367117, −3.09271936415990717055631785082, −3.06501694214300824385375660298, −2.74889795243138222271211633042, −2.71141039109583131863725020263, −2.26767747567647797495396750803, −2.13899149703932707190651415184, −2.08735180492674634089133573452, −1.83230626160597893482175243783, −1.58788031483045122153443687870, −0.978511415818909571962898399012, −0.50771439506466355690325211566,
0.50771439506466355690325211566, 0.978511415818909571962898399012, 1.58788031483045122153443687870, 1.83230626160597893482175243783, 2.08735180492674634089133573452, 2.13899149703932707190651415184, 2.26767747567647797495396750803, 2.71141039109583131863725020263, 2.74889795243138222271211633042, 3.06501694214300824385375660298, 3.09271936415990717055631785082, 3.37643789070114064119387367117, 3.57776446103860760857071855834, 3.61742932060607815507210780877, 3.66816164342689654726684376235, 4.09678938218898410482516246316, 4.34512017051328941468936154739, 4.64730575345348049159149528707, 4.72084020697547006275754002554, 4.90838852424776084526549399267, 4.99231730926276296224480984720, 5.14365787047990759662960074044, 5.35697285724888169395945063147, 5.45112930238953392854414375085, 5.50942299647231440766772225672
Plot not available for L-functions of degree greater than 10.
