| L(s) = 1 | − 2·7-s − 4·13-s + 16-s − 2·19-s + 4·31-s + 2·37-s − 49-s − 4·67-s + 2·73-s + 8·91-s − 2·97-s + 4·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·7-s − 4·13-s + 16-s − 2·19-s + 4·31-s + 2·37-s − 49-s − 4·67-s + 2·73-s + 8·91-s − 2·97-s + 4·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3960217807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3960217807\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2^2$ | \( ( 1 + T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.54956290025715289239451269500, −6.19761420294872753219697435825, −6.00631840992672173824948721282, −5.99795064123008905793119999484, −5.55133829528192910434319213308, −5.52731771031511845794205136028, −5.04387467720188109811182281903, −4.76471905702754943510677871061, −4.71590961534248249581140548895, −4.61354965896856905866341129359, −4.54350670321831489462843601243, −4.13469146767589024855915483654, −4.02414178796792180711046209619, −3.60824206307116348618037432877, −3.18732219738514542657432854343, −3.10337603873073242353598464364, −3.06589816609217972406257089707, −2.66595716561207668830284264304, −2.52278774659984292945981092462, −2.33254566014249200741343549320, −2.17447821830692637158270134735, −1.65455648187943868432493704308, −1.33212768788888504446283124278, −0.77024856201772287650327461074, −0.32571338629952661070556122476,
0.32571338629952661070556122476, 0.77024856201772287650327461074, 1.33212768788888504446283124278, 1.65455648187943868432493704308, 2.17447821830692637158270134735, 2.33254566014249200741343549320, 2.52278774659984292945981092462, 2.66595716561207668830284264304, 3.06589816609217972406257089707, 3.10337603873073242353598464364, 3.18732219738514542657432854343, 3.60824206307116348618037432877, 4.02414178796792180711046209619, 4.13469146767589024855915483654, 4.54350670321831489462843601243, 4.61354965896856905866341129359, 4.71590961534248249581140548895, 4.76471905702754943510677871061, 5.04387467720188109811182281903, 5.52731771031511845794205136028, 5.55133829528192910434319213308, 5.99795064123008905793119999484, 6.00631840992672173824948721282, 6.19761420294872753219697435825, 6.54956290025715289239451269500
