| L(s) = 1 | − 4·11-s + 4·43-s − 4·53-s + 4·67-s + 81-s − 4·107-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 4·11-s + 4·43-s − 4·53-s + 4·67-s + 81-s − 4·107-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2662210483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2662210483\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−6.26208162057582071890118250352, −5.74004286855626282028983112837, −5.62980644947026202755726304271, −5.61527489781115794715216122694, −5.40074117355870009195026048755, −5.32685672007069472138041500314, −4.85005289158952231455538026997, −4.83386695180336351868824776755, −4.67375108574372387670404276091, −4.64943131217025843593529847374, −4.08600791812452125642693982464, −3.86460158089813723806330519505, −3.81428946703503250547807303639, −3.46277645114170231807544365058, −3.37926226028134248985625136621, −2.86033242427510039714820921232, −2.58425928687734926923274575665, −2.53954315053312156510090749268, −2.52095276270662202697734115119, −2.47316934907743895496110657007, −1.79376110546668870700339771282, −1.68873725722883116319785301532, −1.04361465500690467668544771598, −1.02196180870556659317111013430, −0.19612993140155818150179478129,
0.19612993140155818150179478129, 1.02196180870556659317111013430, 1.04361465500690467668544771598, 1.68873725722883116319785301532, 1.79376110546668870700339771282, 2.47316934907743895496110657007, 2.52095276270662202697734115119, 2.53954315053312156510090749268, 2.58425928687734926923274575665, 2.86033242427510039714820921232, 3.37926226028134248985625136621, 3.46277645114170231807544365058, 3.81428946703503250547807303639, 3.86460158089813723806330519505, 4.08600791812452125642693982464, 4.64943131217025843593529847374, 4.67375108574372387670404276091, 4.83386695180336351868824776755, 4.85005289158952231455538026997, 5.32685672007069472138041500314, 5.40074117355870009195026048755, 5.61527489781115794715216122694, 5.62980644947026202755726304271, 5.74004286855626282028983112837, 6.26208162057582071890118250352
