| L(s) = 1 | − 2·2-s − 2·3-s + 5·4-s + 4·6-s − 4·8-s + 3·9-s − 10·12-s + 4·16-s − 12·17-s − 6·18-s + 8·24-s − 2·25-s + 4·29-s + 4·31-s + 22·32-s + 24·34-s + 15·36-s − 16·37-s − 12·41-s − 8·48-s − 12·49-s + 4·50-s + 24·51-s − 8·58-s − 8·62-s − 44·64-s + 16·67-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 5/2·4-s + 1.63·6-s − 1.41·8-s + 9-s − 2.88·12-s + 16-s − 2.91·17-s − 1.41·18-s + 1.63·24-s − 2/5·25-s + 0.742·29-s + 0.718·31-s + 3.88·32-s + 4.11·34-s + 5/2·36-s − 2.63·37-s − 1.87·41-s − 1.15·48-s − 1.71·49-s + 0.565·50-s + 3.36·51-s − 1.05·58-s − 1.01·62-s − 5.5·64-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402809980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402809980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2 T + T^{2} - 4 T^{3} - 11 T^{4} - 4 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
| good | 2 | \( ( 1 + T - T^{2} - 3 T^{3} - T^{4} - 3 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 + 2 T^{2} - 21 T^{4} - 92 T^{6} + 341 T^{8} - 92 p^{2} T^{10} - 21 p^{4} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 + 12 T^{2} + 95 T^{4} + 552 T^{6} + 1969 T^{8} + 552 p^{2} T^{10} + 95 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} \) |
| 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 | \( ( 1 + 6 T + 19 T^{2} + 12 T^{3} - 251 T^{4} + 12 p T^{5} + 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 + 20 T^{2} + 39 T^{4} - 6440 T^{6} - 142879 T^{8} - 6440 p^{2} T^{10} + 39 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 108 p T^{5} - 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T - 27 T^{2} + 116 T^{3} + 605 T^{4} + 116 p T^{5} - 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8 T + 27 T^{2} - 80 T^{3} - 1639 T^{4} - 80 p T^{5} + 27 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 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 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 30 T + 493 T^{2} - 5400 T^{3} + 43069 T^{4} - 5400 p T^{5} + 493 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} )( 1 + 30 T + 493 T^{2} + 5400 T^{3} + 43069 T^{4} + 5400 p T^{5} + 493 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 53 | \( ( 1 - 30 T + 487 T^{2} - 5400 T^{3} + 44749 T^{4} - 5400 p T^{5} + 487 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} )( 1 + 30 T + 487 T^{2} + 5400 T^{3} + 44749 T^{4} + 5400 p T^{5} + 487 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} ) \) |
| 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 | \( 1 + 24 T^{2} - 3145 T^{4} - 164784 T^{6} + 7747729 T^{8} - 164784 p^{2} T^{10} - 3145 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 71 | \( 1 + 134 T^{2} + 12915 T^{4} + 1055116 T^{6} + 76281029 T^{8} + 1055116 p^{2} T^{10} + 12915 p^{4} T^{12} + 134 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 + 144 T^{2} + 15407 T^{4} + 1451232 T^{6} + 126873505 T^{8} + 1451232 p^{2} T^{10} + 15407 p^{4} T^{12} + 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 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 | \( ( 1 - 16 T + 173 T^{2} - 1440 T^{3} + 8681 T^{4} - 1440 p T^{5} + 173 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 - 2 T - 93 T^{2} + 380 T^{3} + 8261 T^{4} + 380 p T^{5} - 93 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−5.10881160582943002626847091896, −4.88023841193362022900817158924, −4.85565252064632847934521381546, −4.68928019924797214950462724040, −4.51846021838089340681344488323, −4.43045537442779878301613375833, −4.34410123473468676769700925314, −4.10805070480544499502035427517, −3.91551972772466679497924925079, −3.68754672046212525200179146429, −3.49407391937860238627529053898, −3.39141934965211634748696968432, −3.09320112531718476959431527089, −3.07931351541466782478857295826, −2.68006223524200454241509391923, −2.59914136017657930514177624396, −2.42365099657029890462795161240, −2.17858318232700489379133078376, −1.84998781243260543389574104742, −1.81696958320895925347014336166, −1.71376286035673470351072787189, −1.57719128515992461880181746263, −0.858793063171722426060353246985, −0.812133292760681154756368517231, −0.44736532433448732277083633163,
0.44736532433448732277083633163, 0.812133292760681154756368517231, 0.858793063171722426060353246985, 1.57719128515992461880181746263, 1.71376286035673470351072787189, 1.81696958320895925347014336166, 1.84998781243260543389574104742, 2.17858318232700489379133078376, 2.42365099657029890462795161240, 2.59914136017657930514177624396, 2.68006223524200454241509391923, 3.07931351541466782478857295826, 3.09320112531718476959431527089, 3.39141934965211634748696968432, 3.49407391937860238627529053898, 3.68754672046212525200179146429, 3.91551972772466679497924925079, 4.10805070480544499502035427517, 4.34410123473468676769700925314, 4.43045537442779878301613375833, 4.51846021838089340681344488323, 4.68928019924797214950462724040, 4.85565252064632847934521381546, 4.88023841193362022900817158924, 5.10881160582943002626847091896
Plot not available for L-functions of degree greater than 10.
