L(s) = 1 | + 8·11-s − 8·19-s + 10·25-s − 16·29-s + 16·31-s + 32·41-s − 2·49-s − 32·61-s − 32·71-s − 16·89-s − 24·109-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 2.41·11-s − 1.83·19-s + 2·25-s − 2.97·29-s + 2.87·31-s + 4.99·41-s − 2/7·49-s − 4.09·61-s − 3.79·71-s − 1.69·89-s − 2.29·109-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1836693796\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1836693796\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7222 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8998 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_4$ | \( ( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 156 T^{2} + 12182 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 36 T^{2} - 538 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 8 T + 14 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 25798 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.32435016968409254855350217916, −6.23023482819972037471261547844, −6.10853163606682044385649771248, −5.92277524434864579312820268897, −5.44393175134077289454331760122, −5.29541849731948431055297063946, −5.14515283166051883873038889939, −4.74989833386195693232903285521, −4.43424640636858183694546440247, −4.37135113917201582958858077952, −4.21733071223587649787686403617, −4.11062186266416378142198170852, −3.86584619562985714261504893541, −3.75491469732468528121836323025, −3.02848833611702520020621631106, −3.02688131945124668336337135387, −2.82512604558852880318999485056, −2.74685062295910314852673802033, −2.16082444557884008439813359293, −2.09964353407427040330629333812, −1.52329529041637748543082846219, −1.40149575098521119209871460294, −1.04021977116270304535962671624, −1.01294888037099936433042195482, −0.06317923134618627038059365220,
0.06317923134618627038059365220, 1.01294888037099936433042195482, 1.04021977116270304535962671624, 1.40149575098521119209871460294, 1.52329529041637748543082846219, 2.09964353407427040330629333812, 2.16082444557884008439813359293, 2.74685062295910314852673802033, 2.82512604558852880318999485056, 3.02688131945124668336337135387, 3.02848833611702520020621631106, 3.75491469732468528121836323025, 3.86584619562985714261504893541, 4.11062186266416378142198170852, 4.21733071223587649787686403617, 4.37135113917201582958858077952, 4.43424640636858183694546440247, 4.74989833386195693232903285521, 5.14515283166051883873038889939, 5.29541849731948431055297063946, 5.44393175134077289454331760122, 5.92277524434864579312820268897, 6.10853163606682044385649771248, 6.23023482819972037471261547844, 6.32435016968409254855350217916