| L(s) = 1 | − 4·2-s + 14·4-s − 48·8-s + 8·9-s − 4·11-s + 129·16-s − 32·18-s + 16·22-s + 12·23-s + 8·25-s + 16·29-s − 336·32-s + 112·36-s + 8·37-s − 12·43-s − 56·44-s − 48·46-s − 32·50-s − 24·53-s − 64·58-s + 834·64-s − 4·67-s − 24·71-s − 384·72-s − 32·74-s − 56·79-s + 34·81-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 7·4-s − 16.9·8-s + 8/3·9-s − 1.20·11-s + 32.2·16-s − 7.54·18-s + 3.41·22-s + 2.50·23-s + 8/5·25-s + 2.97·29-s − 59.3·32-s + 56/3·36-s + 1.31·37-s − 1.82·43-s − 8.44·44-s − 7.07·46-s − 4.52·50-s − 3.29·53-s − 8.40·58-s + 104.·64-s − 0.488·67-s − 2.84·71-s − 45.2·72-s − 3.71·74-s − 6.30·79-s + 34/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4108065093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4108065093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + 20 T^{2} + 231 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 2 | \( ( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 3 | \( ( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 4 T^{2} + 31 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 2 T + 4 T^{2} - 4 p T^{3} - 15 p T^{4} - 4 p^{2} T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 36 T^{2} + 601 T^{4} - 4212 T^{6} + 24960 T^{8} - 4212 p^{2} T^{10} + 601 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 6 T + 4 T^{2} + 84 T^{3} - 333 T^{4} + 84 p T^{5} + 4 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 8 T + 13 T^{2} + 56 T^{3} - 96 T^{4} + 56 p T^{5} + 13 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 4 T - 39 T^{2} + 76 T^{3} + 1064 T^{4} + 76 p T^{5} - 39 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 60 T^{2} + 1201 T^{4} + 57780 T^{6} - 3122160 T^{8} + 57780 p^{2} T^{10} + 1201 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 + 6 T - 36 T^{2} - 84 T^{3} + 1787 T^{4} - 84 p T^{5} - 36 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 - 80 T^{2} + 3726 T^{4} + 343040 T^{6} - 27245485 T^{8} + 343040 p^{2} T^{10} + 3726 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 172 T^{2} + 15873 T^{4} - 1078268 T^{6} + 64063616 T^{8} - 1078268 p^{2} T^{10} + 15873 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 2 T - 108 T^{2} - 44 T^{3} + 7787 T^{4} - 44 p T^{5} - 108 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 100 T^{2} + 127 p T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 14 T + 184 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( 1 + 16 T^{2} - 14178 T^{4} - 22528 T^{6} + 143570339 T^{8} - 22528 p^{2} T^{10} - 14178 p^{4} T^{12} + 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 176 T^{2} + 21567 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.71437182543643194575479475513, −4.63446522518492835427187290604, −4.38238361967451160998811537737, −4.01453366613151528631089594806, −3.97218738204525399490237577974, −3.93042181009860201554773652978, −3.57910202067471369815569415453, −3.18876440264667660329093342746, −3.17073437718250360058143175614, −3.15993074784555366260208913196, −3.03905272948381252521085574597, −2.99003773010858624601410018975, −2.82150506863908742318423780555, −2.65024334726365737682228740939, −2.61430168286375432095907888604, −2.46150903536508792741100307748, −2.05312526508072952704541987436, −1.88022673012806132182412890956, −1.83045509365056935015039331927, −1.45609047296740185031699039074, −1.38600325544799745284109894002, −1.14423458857019977662047049032, −0.977760354114844897107363975100, −0.53232870906208110014845274271, −0.18581199578397290421494502076,
0.18581199578397290421494502076, 0.53232870906208110014845274271, 0.977760354114844897107363975100, 1.14423458857019977662047049032, 1.38600325544799745284109894002, 1.45609047296740185031699039074, 1.83045509365056935015039331927, 1.88022673012806132182412890956, 2.05312526508072952704541987436, 2.46150903536508792741100307748, 2.61430168286375432095907888604, 2.65024334726365737682228740939, 2.82150506863908742318423780555, 2.99003773010858624601410018975, 3.03905272948381252521085574597, 3.15993074784555366260208913196, 3.17073437718250360058143175614, 3.18876440264667660329093342746, 3.57910202067471369815569415453, 3.93042181009860201554773652978, 3.97218738204525399490237577974, 4.01453366613151528631089594806, 4.38238361967451160998811537737, 4.63446522518492835427187290604, 4.71437182543643194575479475513
Plot not available for L-functions of degree greater than 10.
