| L(s) = 1 | + 8·2-s + 32·4-s + 80·8-s − 8·11-s + 120·16-s − 64·22-s − 8·25-s + 32·32-s + 32·37-s − 256·44-s − 64·50-s − 384·64-s − 88·71-s + 256·74-s − 18·81-s − 640·88-s − 256·100-s + 32·121-s + 127-s − 1.21e3·128-s + 131-s + 137-s + 139-s − 704·142-s + 1.02e3·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 16·4-s + 28.2·8-s − 2.41·11-s + 30·16-s − 13.6·22-s − 8/5·25-s + 5.65·32-s + 5.26·37-s − 38.5·44-s − 9.05·50-s − 48·64-s − 10.4·71-s + 29.7·74-s − 2·81-s − 68.2·88-s − 25.5·100-s + 2.90·121-s + 0.0887·127-s − 107.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 59.0·142-s + 84.1·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1892248053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1892248053\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T + p T^{2} )^{4} \) |
| 3 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| good | 11 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2}( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 - 622 T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 + 578 T^{4} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T + p T^{2} )^{4}( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + p T^{2} )^{8} \) |
| 83 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 172 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + p^{2} T^{4} )^{4} \) |
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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
−4.83807021041435852017921887939, −4.77863514302186542416004835980, −4.75857343196859391266821842017, −4.45245232745507942791520447937, −4.24336692929180638930242315723, −4.22437804819369063537733802283, −4.13377498661812420473173876562, −4.12515787919215949475127097430, −4.11112400666195571737306843163, −3.91904549604205134512765743198, −3.37995418874857946200314010426, −3.24108326646807931099691494424, −3.12826777314955601810702040993, −3.10630673714514600321804324155, −2.88928710142080366712238787011, −2.88474022225834489232553918038, −2.79823063800433575814856323788, −2.48650468026176066059218592862, −2.35456984949009682709533035311, −2.14228843057655517316536864041, −1.99734880895895489977427601310, −1.55346847005490598676258615897, −1.55291088692209992717855079030, −0.819407344400545017162564550988, −0.02451729550196514022643291301,
0.02451729550196514022643291301, 0.819407344400545017162564550988, 1.55291088692209992717855079030, 1.55346847005490598676258615897, 1.99734880895895489977427601310, 2.14228843057655517316536864041, 2.35456984949009682709533035311, 2.48650468026176066059218592862, 2.79823063800433575814856323788, 2.88474022225834489232553918038, 2.88928710142080366712238787011, 3.10630673714514600321804324155, 3.12826777314955601810702040993, 3.24108326646807931099691494424, 3.37995418874857946200314010426, 3.91904549604205134512765743198, 4.11112400666195571737306843163, 4.12515787919215949475127097430, 4.13377498661812420473173876562, 4.22437804819369063537733802283, 4.24336692929180638930242315723, 4.45245232745507942791520447937, 4.75857343196859391266821842017, 4.77863514302186542416004835980, 4.83807021041435852017921887939
Plot not available for L-functions of degree greater than 10.
