| L(s) = 1 | − 5·9-s − 32·13-s + 2·25-s + 4·37-s − 4·49-s − 12·61-s − 28·73-s + 9·81-s + 64·97-s − 20·109-s + 160·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 472·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 5/3·9-s − 8.87·13-s + 2/5·25-s + 0.657·37-s − 4/7·49-s − 1.53·61-s − 3.27·73-s + 81-s + 6.49·97-s − 1.91·109-s + 14.7·117-s + 2·121-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 + 36.3·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1423760303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1423760303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 17 | \( ( 1 + 23 T^{2} + 240 T^{4} + 23 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 - 35 T^{2} + 696 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 53 T^{2} + 1848 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 11 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 5 T^{2} - 2184 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 95 T^{2} + 6216 T^{4} + 95 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 107 T^{2} + 7968 T^{4} - 107 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 53 T^{2} - 1680 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 17 T + p T^{2} )^{4} \) |
| 79 | \( ( 1 + 77 T^{2} - 312 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 167 T^{2} + 19968 T^{4} + 167 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
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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
−4.99156857953880120483465870089, −4.97649382645897570858202002128, −4.86863164848196438843490050539, −4.77634433334783461924630185088, −4.64769253056344577794311766035, −4.58162178697617460034654886409, −4.52341667919293087115619729582, −4.26135540380764316530326904408, −3.91587627255252472078617711815, −3.88163342559773857486782688099, −3.58613002633388467907249154557, −3.19127376054444291430970359704, −3.16053331028231878747144342697, −2.93379699259229145443724214264, −2.87939657310962762510604170159, −2.77919378883500755212716798197, −2.54750931667252002512664237158, −2.43556928282328463914073216731, −2.15055267506927584638255289964, −2.12536284104950622157825801596, −1.81433508411473151668280114053, −1.79182818441912778562778588356, −0.985192418375809989861930428334, −0.42318459398530991891274617564, −0.17256259048464793447344690168,
0.17256259048464793447344690168, 0.42318459398530991891274617564, 0.985192418375809989861930428334, 1.79182818441912778562778588356, 1.81433508411473151668280114053, 2.12536284104950622157825801596, 2.15055267506927584638255289964, 2.43556928282328463914073216731, 2.54750931667252002512664237158, 2.77919378883500755212716798197, 2.87939657310962762510604170159, 2.93379699259229145443724214264, 3.16053331028231878747144342697, 3.19127376054444291430970359704, 3.58613002633388467907249154557, 3.88163342559773857486782688099, 3.91587627255252472078617711815, 4.26135540380764316530326904408, 4.52341667919293087115619729582, 4.58162178697617460034654886409, 4.64769253056344577794311766035, 4.77634433334783461924630185088, 4.86863164848196438843490050539, 4.97649382645897570858202002128, 4.99156857953880120483465870089
Plot not available for L-functions of degree greater than 10.
