L(s) = 1 | − 5-s + 4·11-s − 2·13-s + 2·17-s − 4·19-s − 8·23-s + 25-s − 2·29-s − 6·37-s − 6·41-s − 4·43-s − 10·53-s − 4·55-s + 12·59-s + 14·61-s + 2·65-s − 12·67-s − 8·71-s − 10·73-s − 16·79-s − 12·83-s − 2·85-s + 10·89-s + 4·95-s − 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.986·37-s − 0.937·41-s − 0.609·43-s − 1.37·53-s − 0.539·55-s + 1.56·59-s + 1.79·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s − 1.17·73-s − 1.80·79-s − 1.31·83-s − 0.216·85-s + 1.05·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5319180561\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5319180561\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.27728677232926, −13.00401110101837, −12.20435562637006, −11.98570062594531, −11.68432068242884, −11.08442127523276, −10.42133487777372, −10.04313487453180, −9.627467885487323, −8.991486690116313, −8.437881398074515, −8.192903156148241, −7.498699548009983, −6.886852821185117, −6.679624206938428, −5.890721537257988, −5.532605340397766, −4.763554478104056, −4.167993610859639, −3.883755792115889, −3.239345418061674, −2.545880376491199, −1.742760461030147, −1.389742226569904, −0.2114165766652883,
0.2114165766652883, 1.389742226569904, 1.742760461030147, 2.545880376491199, 3.239345418061674, 3.883755792115889, 4.167993610859639, 4.763554478104056, 5.532605340397766, 5.890721537257988, 6.679624206938428, 6.886852821185117, 7.498699548009983, 8.192903156148241, 8.437881398074515, 8.991486690116313, 9.627467885487323, 10.04313487453180, 10.42133487777372, 11.08442127523276, 11.68432068242884, 11.98570062594531, 12.20435562637006, 13.00401110101837, 13.27728677232926