| L(s) = 1 | + 64·17-s + 1.27e3·25-s + 1.85e3·41-s + 4.42e3·49-s + 3.59e3·73-s + 2.82e4·89-s − 9.46e3·97-s + 6.23e4·113-s − 1.93e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.46e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 0.221·17-s + 2.04·25-s + 1.10·41-s + 1.84·49-s + 0.674·73-s + 3.57·89-s − 1.00·97-s + 4.88·113-s − 1.32·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.31·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(13.93194769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.93194769\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 638 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 2210 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 9698 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 47330 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 52702 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 103870 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1237694 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1408994 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2371678 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 464 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 6524258 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3787394 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5596670 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10995938 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 9586274 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 8967842 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 40206530 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 898 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 3870050 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 94740386 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 7072 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2366 T + p^{4} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.57794150112910137953158300536, −5.94473413549739971727819287303, −5.93837423830918592002370043248, −5.83431673925106516963460771485, −5.81880219400721396186704360350, −5.11929040834571180651963432325, −4.97704966326612309142049178949, −4.88113683510336661385880493796, −4.82753866541667502806060471092, −4.31194476928256872328301443237, −4.10323369476065649740282737796, −3.84595758084896894922877514792, −3.80140304344882096964392359648, −3.19643649703433340095535988410, −3.05877987032950407806167101332, −2.96057282596189563932174992366, −2.72408432840358126403712821439, −2.06876912991547478299364811721, −2.01955753413185021238281899945, −1.99377752078594314505770008129, −1.34549808876081211947137105274, −1.06755717346053636301011323110, −0.62362980995039627544022645706, −0.57051986389146157631598174003, −0.51436029191417336001487739364,
0.51436029191417336001487739364, 0.57051986389146157631598174003, 0.62362980995039627544022645706, 1.06755717346053636301011323110, 1.34549808876081211947137105274, 1.99377752078594314505770008129, 2.01955753413185021238281899945, 2.06876912991547478299364811721, 2.72408432840358126403712821439, 2.96057282596189563932174992366, 3.05877987032950407806167101332, 3.19643649703433340095535988410, 3.80140304344882096964392359648, 3.84595758084896894922877514792, 4.10323369476065649740282737796, 4.31194476928256872328301443237, 4.82753866541667502806060471092, 4.88113683510336661385880493796, 4.97704966326612309142049178949, 5.11929040834571180651963432325, 5.81880219400721396186704360350, 5.83431673925106516963460771485, 5.93837423830918592002370043248, 5.94473413549739971727819287303, 6.57794150112910137953158300536
