| L(s) = 1 | + 4·19-s − 4·31-s + 4·37-s − 4·67-s + 4·79-s − 4·97-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 + 233-s + ⋯ |
| L(s) = 1 | + 4·19-s − 4·31-s + 4·37-s − 4·67-s + 4·79-s − 4·97-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 + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.210449760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.210449760\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 41 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
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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
−7.23426794969014933045527138196, −6.74923924152430453369298963805, −6.45938969197599088029296484244, −6.44557989041099751514526528077, −6.05447915068706497989652631502, −5.76069325817840408685751921537, −5.74252781928579290624026594715, −5.39148234811469888807271572053, −5.31785870282559752922319429936, −5.10386941311006273532775149205, −4.90586132689598299368643665073, −4.49882591202135385695074938502, −4.34282223332171369858525347135, −4.01180857281692616062435471434, −3.63999250369576898041391731541, −3.60932274942734157317381646821, −3.46963647465725445458914057321, −3.01552189197917663611260438185, −2.62340723348072100197295769917, −2.56511491309463017486591582628, −2.48558848672711717445975484500, −1.57003159426579509028012352298, −1.44206010273922956149895223690, −1.39103659892376642828207477801, −0.72336711461650633457673066807,
0.72336711461650633457673066807, 1.39103659892376642828207477801, 1.44206010273922956149895223690, 1.57003159426579509028012352298, 2.48558848672711717445975484500, 2.56511491309463017486591582628, 2.62340723348072100197295769917, 3.01552189197917663611260438185, 3.46963647465725445458914057321, 3.60932274942734157317381646821, 3.63999250369576898041391731541, 4.01180857281692616062435471434, 4.34282223332171369858525347135, 4.49882591202135385695074938502, 4.90586132689598299368643665073, 5.10386941311006273532775149205, 5.31785870282559752922319429936, 5.39148234811469888807271572053, 5.74252781928579290624026594715, 5.76069325817840408685751921537, 6.05447915068706497989652631502, 6.44557989041099751514526528077, 6.45938969197599088029296484244, 6.74923924152430453369298963805, 7.23426794969014933045527138196
