| L(s) = 1 | + 2·9-s + 25-s − 8·29-s + 81-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + 227-s + ⋯ |
| L(s) = 1 | + 2·9-s + 25-s − 8·29-s + 81-s − 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 2·225-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7546168170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7546168170\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{8} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.13807167678287230776968203034, −6.04344752980260771131176227772, −5.79345676558268074906752727867, −5.44023125587389568714824232883, −5.37285176903902925405164078577, −5.23526744417419994762963215355, −5.04314864506762778018709152836, −4.96227370926182661166792839586, −4.38458918026463373274329170753, −4.34068765620422180521038197507, −3.98058925832195806117775795481, −3.96959647685308557387790409369, −3.80565058961165716497140631245, −3.78714565461400642748245527060, −3.30912613006016697680267459622, −3.10337384369007198507850649765, −2.95879098209780670412631301507, −2.50630904097057243613620254514, −2.16833957810339294480032665375, −2.03367807888651183556892160943, −1.81962186789723008512524606350, −1.59431164987275454556467023760, −1.33107996145596575114132561806, −1.11831237650297407011394469092, −0.28549954460240961735295879634,
0.28549954460240961735295879634, 1.11831237650297407011394469092, 1.33107996145596575114132561806, 1.59431164987275454556467023760, 1.81962186789723008512524606350, 2.03367807888651183556892160943, 2.16833957810339294480032665375, 2.50630904097057243613620254514, 2.95879098209780670412631301507, 3.10337384369007198507850649765, 3.30912613006016697680267459622, 3.78714565461400642748245527060, 3.80565058961165716497140631245, 3.96959647685308557387790409369, 3.98058925832195806117775795481, 4.34068765620422180521038197507, 4.38458918026463373274329170753, 4.96227370926182661166792839586, 5.04314864506762778018709152836, 5.23526744417419994762963215355, 5.37285176903902925405164078577, 5.44023125587389568714824232883, 5.79345676558268074906752727867, 6.04344752980260771131176227772, 6.13807167678287230776968203034
