L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.654 − 0.755i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (0.841 − 0.540i)6-s + (−0.142 + 0.989i)7-s + (0.415 − 0.909i)8-s + (−0.142 + 0.989i)9-s + (−0.654 + 0.755i)10-s + (0.841 + 0.540i)11-s + (0.415 + 0.909i)12-s + (0.415 + 0.909i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)17-s + ⋯ |
L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.654 − 0.755i)3-s + (−0.959 − 0.281i)4-s + (0.841 + 0.540i)5-s + (0.841 − 0.540i)6-s + (−0.142 + 0.989i)7-s + (0.415 − 0.909i)8-s + (−0.142 + 0.989i)9-s + (−0.654 + 0.755i)10-s + (0.841 + 0.540i)11-s + (0.415 + 0.909i)12-s + (0.415 + 0.909i)13-s + (−0.959 − 0.281i)14-s + (−0.142 − 0.989i)15-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0678 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0678 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5287150598 + 0.4939567843i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5287150598 + 0.4939567843i\) |
\(L(1)\) |
\(\approx\) |
\(0.7250076969 + 0.3846948307i\) |
\(L(1)\) |
\(\approx\) |
\(0.7250076969 + 0.3846948307i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (-0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 11 | \( 1 + (0.841 + 0.540i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (-0.959 + 0.281i)T \) |
| 19 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.654 - 0.755i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (0.415 - 0.909i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.959 + 0.281i)T \) |
| 43 | \( 1 + (-0.959 + 0.281i)T \) |
| 47 | \( 1 + (-0.654 - 0.755i)T \) |
| 53 | \( 1 + (-0.959 - 0.281i)T \) |
| 59 | \( 1 + (0.415 - 0.909i)T \) |
| 61 | \( 1 + (0.841 - 0.540i)T \) |
| 71 | \( 1 + (-0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.841 - 0.540i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (0.841 + 0.540i)T \) |
| 89 | \( 1 + (-0.654 + 0.755i)T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.041072907353938766663169516657, −30.22931027850929801347934023355, −29.39378906460510106221438071060, −28.65140431525968510354482656262, −27.43891556981005184519687986113, −26.80900489936403573910237369548, −25.36564206549339990592749330833, −23.617586930593237641714420695516, −22.52784943943369997094050246806, −21.65061407283893246492470236593, −20.614051302128333442401090415063, −19.83069032362391753228906230511, −17.94179267024243624904041313714, −17.24151824426061613921031744920, −16.217705285324934570056258694595, −14.19808776471706635371952430286, −13.16032982271417220036855616126, −11.80571138596262074446424675142, −10.560447979386750470996791201372, −9.81526168598399783790040725009, −8.561122386125019388686072587878, −6.14220077986886792201507628855, −4.71189507750891936266818758080, −3.49692364855410980533804181845, −1.1470664372553927713297356992,
2.00681134059704995185373529770, 4.79539015325641686533966236811, 6.41739294707983558165438022962, 6.56356146027812669108332207971, 8.51357026249651207638058596946, 9.73699394514464922850335963066, 11.44613638764016576751858991950, 12.92403902122432943690314847637, 13.96891143548023203186432454746, 15.18171961414142079390451918784, 16.60517632383947675368290834212, 17.68462603104157829865659546397, 18.33490326790241230400662315053, 19.39438961538534068482371114188, 21.87391783467585768130984475541, 22.28822479312421072110172262409, 23.63073716801301144712335503379, 24.725583861779717770274176848304, 25.398051848999565923274670482640, 26.49058798435365789341102460593, 28.19188200412800049751010628856, 28.632479595607787828151017628892, 30.2573048520866359585197508666, 31.095478345399793587601800786848, 32.66122837653272190138685206967