L(s) = 1 | − 5-s + 11-s − 6·13-s − 3·17-s + 19-s + 23-s + 25-s − 5·29-s − 7·37-s + 10·41-s − 13·43-s + 7·47-s + 10·53-s − 55-s − 13·59-s − 8·61-s + 6·65-s − 14·67-s − 5·71-s + 6·73-s + 12·79-s + 12·83-s + 3·85-s − 16·89-s − 95-s + 17·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.66·13-s − 0.727·17-s + 0.229·19-s + 0.208·23-s + 1/5·25-s − 0.928·29-s − 1.15·37-s + 1.56·41-s − 1.98·43-s + 1.02·47-s + 1.37·53-s − 0.134·55-s − 1.69·59-s − 1.02·61-s + 0.744·65-s − 1.71·67-s − 0.593·71-s + 0.702·73-s + 1.35·79-s + 1.31·83-s + 0.325·85-s − 1.69·89-s − 0.102·95-s + 1.72·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.98980928193265, −13.58192421361254, −13.04677789097483, −12.41826047607475, −12.05453656563505, −11.77774242495822, −10.96749670028897, −10.74115571680522, −10.00741522152290, −9.619949470009415, −8.911238055280119, −8.793637306692108, −7.859297332262952, −7.394451770438219, −7.191227567493379, −6.468420172590194, −5.878594641905269, −5.237654715727442, −4.636946986229994, −4.360965414333981, −3.468280737129365, −3.062900377015553, −2.208090124795037, −1.802174765327804, −0.7110059450643951, 0,
0.7110059450643951, 1.802174765327804, 2.208090124795037, 3.062900377015553, 3.468280737129365, 4.360965414333981, 4.636946986229994, 5.237654715727442, 5.878594641905269, 6.468420172590194, 7.191227567493379, 7.394451770438219, 7.859297332262952, 8.793637306692108, 8.911238055280119, 9.619949470009415, 10.00741522152290, 10.74115571680522, 10.96749670028897, 11.77774242495822, 12.05453656563505, 12.41826047607475, 13.04677789097483, 13.58192421361254, 13.98980928193265