| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s + 3·9-s + 2·12-s + 6·18-s − 8·25-s − 12·29-s + 8·31-s − 2·32-s + 3·36-s − 4·37-s + 12·41-s + 4·49-s − 16·50-s − 24·58-s + 16·62-s − 4·64-s − 32·67-s − 8·74-s − 16·75-s + 24·82-s + 24·83-s − 24·87-s + 16·93-s − 4·96-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 9-s + 0.577·12-s + 1.41·18-s − 8/5·25-s − 2.22·29-s + 1.43·31-s − 0.353·32-s + 1/2·36-s − 0.657·37-s + 1.87·41-s + 4/7·49-s − 2.26·50-s − 3.15·58-s + 2.03·62-s − 1/2·64-s − 3.90·67-s − 0.929·74-s − 1.84·75-s + 2.65·82-s + 2.63·83-s − 2.57·87-s + 1.65·93-s − 0.408·96-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.15444774\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.15444774\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 3 | \( 1 - 2 T + T^{2} + 4 T^{3} - 11 T^{4} + 4 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 8 T^{2} + 39 T^{4} + 112 T^{6} - 79 T^{8} + 112 p^{2} T^{10} + 39 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 - 4 T^{2} - 33 T^{4} + 328 T^{6} + 305 T^{8} + 328 p^{2} T^{10} - 33 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 + 8 T^{2} - 105 T^{4} - 2192 T^{6} + 209 T^{8} - 2192 p^{2} T^{10} - 105 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 132 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} + 184 p T^{5} - 15 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T - 33 T^{2} - 140 T^{3} + 941 T^{4} - 140 p T^{5} - 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 6 T - 5 T^{2} + 276 T^{3} - 1451 T^{4} + 276 p T^{5} - 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 47 | \( 1 - 4 T^{2} - 2193 T^{4} + 17608 T^{6} + 4773905 T^{8} + 17608 p^{2} T^{10} - 2193 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 56 T^{2} + 327 T^{4} - 138992 T^{6} - 8702095 T^{8} - 138992 p^{2} T^{10} + 327 p^{4} T^{12} + 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 10 T^{2} - 3381 T^{4} + 68620 T^{6} + 11083061 T^{8} + 68620 p^{2} T^{10} - 3381 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 + 104 T^{2} + 7095 T^{4} + 350896 T^{6} + 10092689 T^{8} + 350896 p^{2} T^{10} + 7095 p^{4} T^{12} + 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 71 | \( 1 + 92 T^{2} + 3423 T^{4} - 148856 T^{6} - 30950095 T^{8} - 148856 p^{2} T^{10} + 3423 p^{4} T^{12} + 92 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 + 140 T^{2} + 13359 T^{4} + 996520 T^{6} + 56139281 T^{8} + 996520 p^{2} T^{10} + 13359 p^{4} T^{12} + 140 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 12 T + 61 T^{2} + 264 T^{3} - 8231 T^{4} + 264 p T^{5} + 61 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 18 T + p T^{2} )^{4}( 1 + 18 T + p T^{2} )^{4} \) |
| 97 | \( ( 1 + 8 T - 33 T^{2} - 1040 T^{3} - 5119 T^{4} - 1040 p T^{5} - 33 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.44820920383551541197257982718, −4.37902729172947846918784674194, −4.21213128572954052212855529603, −4.10608792316408735246493287198, −4.06522347398531182473794523109, −3.97573114009837876083417944852, −3.56352516336243622037379900636, −3.53917007888652740246784519222, −3.43541732870481485847402827793, −3.19166570447383513359323279890, −3.16811680070821641828285700681, −3.12293629647016340725159040479, −3.07200540731145320288242185081, −2.51892379268742147737487771126, −2.51575331104782293035236360881, −2.30404016990581000897500395362, −2.10975783457038230183857160864, −2.06488182160869593893861182237, −1.86655346657019570840673722986, −1.82602018325612161007012466836, −1.36767913838784326792946203070, −1.30359000900874366050399778144, −0.974897062890711841160876706081, −0.55802558841941752156895039595, −0.33445224909518010208599312905,
0.33445224909518010208599312905, 0.55802558841941752156895039595, 0.974897062890711841160876706081, 1.30359000900874366050399778144, 1.36767913838784326792946203070, 1.82602018325612161007012466836, 1.86655346657019570840673722986, 2.06488182160869593893861182237, 2.10975783457038230183857160864, 2.30404016990581000897500395362, 2.51575331104782293035236360881, 2.51892379268742147737487771126, 3.07200540731145320288242185081, 3.12293629647016340725159040479, 3.16811680070821641828285700681, 3.19166570447383513359323279890, 3.43541732870481485847402827793, 3.53917007888652740246784519222, 3.56352516336243622037379900636, 3.97573114009837876083417944852, 4.06522347398531182473794523109, 4.10608792316408735246493287198, 4.21213128572954052212855529603, 4.37902729172947846918784674194, 4.44820920383551541197257982718
Plot not available for L-functions of degree greater than 10.
