L(s) = 1 | + 4·5-s − 6·11-s + 5·13-s + 2·17-s − 19-s − 6·23-s + 11·25-s − 3·31-s − 3·37-s − 6·41-s + 5·43-s + 4·47-s − 6·53-s − 24·55-s − 6·59-s − 2·61-s + 20·65-s + 7·67-s + 16·71-s + 3·73-s − 11·79-s + 12·83-s + 8·85-s + 4·89-s − 4·95-s + 6·97-s + 101-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 1.80·11-s + 1.38·13-s + 0.485·17-s − 0.229·19-s − 1.25·23-s + 11/5·25-s − 0.538·31-s − 0.493·37-s − 0.937·41-s + 0.762·43-s + 0.583·47-s − 0.824·53-s − 3.23·55-s − 0.781·59-s − 0.256·61-s + 2.48·65-s + 0.855·67-s + 1.89·71-s + 0.351·73-s − 1.23·79-s + 1.31·83-s + 0.867·85-s + 0.423·89-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.063222405\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.063222405\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.33436458397161, −14.50625026237463, −13.91339758013874, −13.70241410924312, −13.20360475581434, −12.65812603371468, −12.27342126106673, −11.18963023458783, −10.78255996935006, −10.32938524951196, −9.892219033092616, −9.326425712776560, −8.661132792693706, −8.135279033709756, −7.578799106811110, −6.666642809904482, −6.181278087491135, −5.569886208691699, −5.358642436382156, −4.531515599216150, −3.556182730317107, −2.921062865399233, −2.101277154786963, −1.741569093348391, −0.6593400871088454,
0.6593400871088454, 1.741569093348391, 2.101277154786963, 2.921062865399233, 3.556182730317107, 4.531515599216150, 5.358642436382156, 5.569886208691699, 6.181278087491135, 6.666642809904482, 7.578799106811110, 8.135279033709756, 8.661132792693706, 9.326425712776560, 9.892219033092616, 10.32938524951196, 10.78255996935006, 11.18963023458783, 12.27342126106673, 12.65812603371468, 13.20360475581434, 13.70241410924312, 13.91339758013874, 14.50625026237463, 15.33436458397161