| L(s) = 1 | − 2·4-s + 3·16-s + 4·37-s + 8·43-s − 6·64-s − 4·67-s + 4·79-s + 81-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s + 4·37-s + 8·43-s − 6·64-s − 4·67-s + 4·79-s + 81-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.065998890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065998890\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T^{4} + T^{8} \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | \( ( 1 + T^{4} )^{4} \) |
| 17 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 19 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 23 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 37 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
| 41 | \( ( 1 + T^{4} )^{4} \) |
| 43 | \( ( 1 - T + T^{2} )^{8} \) |
| 47 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 53 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 59 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 61 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 67 | \( ( 1 + T )^{8}( 1 - T + T^{2} )^{4} \) |
| 71 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 73 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 79 | \( ( 1 - T )^{8}( 1 + T + T^{2} )^{4} \) |
| 83 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 89 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{2} )^{8} \) |
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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
−3.80561279544192694776037063801, −3.66883353951777889530017244843, −3.53707602865009621540617963234, −3.37954727658765970461432211318, −3.36381879851074488074806298224, −3.24510002881566207448555937634, −3.03665542293979162554348419371, −2.85705967726781160335234959589, −2.67334103852974775078206535549, −2.64696323198864025968511512521, −2.62241871045909889029888808762, −2.45490875133514274771124305457, −2.41555970042375513141833003728, −2.37971691273080649180534669148, −2.11819952715161250961236643257, −1.96906754444463150875983252027, −1.63459115420987412385289169551, −1.62430878052678876931228914385, −1.30758622855200613069442331936, −1.25473568343187657848978236910, −1.12305164762050886129151185129, −0.914992692965153432024391057230, −0.845519612276539822270543419409, −0.78493224780645069490074270088, −0.30601491013005029252062701626,
0.30601491013005029252062701626, 0.78493224780645069490074270088, 0.845519612276539822270543419409, 0.914992692965153432024391057230, 1.12305164762050886129151185129, 1.25473568343187657848978236910, 1.30758622855200613069442331936, 1.62430878052678876931228914385, 1.63459115420987412385289169551, 1.96906754444463150875983252027, 2.11819952715161250961236643257, 2.37971691273080649180534669148, 2.41555970042375513141833003728, 2.45490875133514274771124305457, 2.62241871045909889029888808762, 2.64696323198864025968511512521, 2.67334103852974775078206535549, 2.85705967726781160335234959589, 3.03665542293979162554348419371, 3.24510002881566207448555937634, 3.36381879851074488074806298224, 3.37954727658765970461432211318, 3.53707602865009621540617963234, 3.66883353951777889530017244843, 3.80561279544192694776037063801
Plot not available for L-functions of degree greater than 10.
