| L(s) = 1 | − 12·9-s − 28·49-s + 90·81-s − 80·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 4·9-s − 4·49-s + 10·81-s − 7.66·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2956637584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2956637584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p T^{2} )^{4} \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 34 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 - p T^{2} )^{8} \) |
| 17 | \( ( 1 - 178 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 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 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 1262 T^{4} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2}( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | \( ( 1 - p T^{2} )^{8} \) |
| 79 | \( ( 1 + p T^{2} )^{8} \) |
| 83 | \( ( 1 + p T^{2} )^{8} \) |
| 89 | \( ( 1 + 13742 T^{4} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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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
−4.53871779734137336448874358864, −4.34307491392386193473027543849, −4.32954546392788019038810049104, −4.15158748298575492094282798559, −4.04672656385123476953049451924, −3.99373225421696653832381262531, −3.57903812301968250769548692205, −3.38944931563854722287913813750, −3.37973695673668757480552524471, −3.26442526198933076974934335948, −3.23744750725891920203550189196, −2.99495342933142430903112229640, −2.75565937701162677545875938018, −2.71533331535696133139851638364, −2.68063384623106692086802315039, −2.43822872264189578487477621655, −2.12453580861952486796448052438, −2.07502265437936819412105566243, −1.84390936584989219526270869446, −1.66558862120719017410085905170, −1.39258147639189917845121122488, −1.11857736872093081676544675762, −0.852988098913201820327948509097, −0.36859924063577907253118153996, −0.14498208551501553164315421097,
0.14498208551501553164315421097, 0.36859924063577907253118153996, 0.852988098913201820327948509097, 1.11857736872093081676544675762, 1.39258147639189917845121122488, 1.66558862120719017410085905170, 1.84390936584989219526270869446, 2.07502265437936819412105566243, 2.12453580861952486796448052438, 2.43822872264189578487477621655, 2.68063384623106692086802315039, 2.71533331535696133139851638364, 2.75565937701162677545875938018, 2.99495342933142430903112229640, 3.23744750725891920203550189196, 3.26442526198933076974934335948, 3.37973695673668757480552524471, 3.38944931563854722287913813750, 3.57903812301968250769548692205, 3.99373225421696653832381262531, 4.04672656385123476953049451924, 4.15158748298575492094282798559, 4.32954546392788019038810049104, 4.34307491392386193473027543849, 4.53871779734137336448874358864
Plot not available for L-functions of degree greater than 10.
