| L(s) = 1 | − 2·25-s + 8·29-s − 2·81-s − 4·121-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 + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 2·25-s + 8·29-s − 2·81-s − 4·121-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 + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.067198029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067198029\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_1$ | \( ( 1 - T )^{8} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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
−7.35451151893149358426022803706, −6.81632344585498536511998192800, −6.69537744581888639462228010301, −6.67544739109877215453841978458, −6.31935010476225589123642825744, −6.30287041845908139821184006248, −6.01752856155014287296333372906, −5.62460238959048229618598604484, −5.57737949411085214059723978377, −5.04475919863238095014356382329, −4.93017583447507328552005666325, −4.81691452895326005921719859864, −4.45278513780856770361299141264, −4.23054684244487218955244404998, −4.22832342259436788253194371123, −3.66937449300907883328296566916, −3.54039684525809606420512015719, −2.95428087845050544134746477136, −2.89709334595176181591152831215, −2.74726002137464069677077475211, −2.33808524479508791222619385669, −2.18861563401523290385669901959, −1.35849802328338712209255188712, −1.29529814459850154219268865617, −0.887202592886606159002293330930,
0.887202592886606159002293330930, 1.29529814459850154219268865617, 1.35849802328338712209255188712, 2.18861563401523290385669901959, 2.33808524479508791222619385669, 2.74726002137464069677077475211, 2.89709334595176181591152831215, 2.95428087845050544134746477136, 3.54039684525809606420512015719, 3.66937449300907883328296566916, 4.22832342259436788253194371123, 4.23054684244487218955244404998, 4.45278513780856770361299141264, 4.81691452895326005921719859864, 4.93017583447507328552005666325, 5.04475919863238095014356382329, 5.57737949411085214059723978377, 5.62460238959048229618598604484, 6.01752856155014287296333372906, 6.30287041845908139821184006248, 6.31935010476225589123642825744, 6.67544739109877215453841978458, 6.69537744581888639462228010301, 6.81632344585498536511998192800, 7.35451151893149358426022803706
