L(s) = 1 | + (−0.900 − 0.433i)4-s + (−0.733 − 0.680i)7-s + (−1.88 + 0.582i)13-s + (0.623 + 0.781i)16-s + (0.222 + 0.385i)19-s + (−0.733 + 0.680i)25-s + (0.365 + 0.930i)28-s + 1.65·31-s + (1.36 + 0.930i)37-s + (1.44 − 0.218i)43-s + (0.0747 + 0.997i)49-s + (1.95 + 0.294i)52-s + (−0.134 + 0.0648i)61-s + (−0.222 − 0.974i)64-s + 0.730·67-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)4-s + (−0.733 − 0.680i)7-s + (−1.88 + 0.582i)13-s + (0.623 + 0.781i)16-s + (0.222 + 0.385i)19-s + (−0.733 + 0.680i)25-s + (0.365 + 0.930i)28-s + 1.65·31-s + (1.36 + 0.930i)37-s + (1.44 − 0.218i)43-s + (0.0747 + 0.997i)49-s + (1.95 + 0.294i)52-s + (−0.134 + 0.0648i)61-s + (−0.222 − 0.974i)64-s + 0.730·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6585009959\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6585009959\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.733 + 0.680i)T \) |
good | 2 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 5 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 11 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 13 | \( 1 + (1.88 - 0.582i)T + (0.826 - 0.563i)T^{2} \) |
| 17 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 19 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 31 | \( 1 - 1.65T + T^{2} \) |
| 37 | \( 1 + (-1.36 - 0.930i)T + (0.365 + 0.930i)T^{2} \) |
| 41 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 0.218i)T + (0.955 - 0.294i)T^{2} \) |
| 47 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 59 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 61 | \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 - 0.730T + T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (1.72 + 0.531i)T + (0.826 + 0.563i)T^{2} \) |
| 79 | \( 1 - 0.149T + T^{2} \) |
| 83 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 89 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 97 | \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.897746560270578670220301141329, −7.84414948927733129577888525763, −7.38015917477939998914327831035, −6.44151499162125857060873929320, −5.77100984853167728117966741393, −4.73338849065058476795574324906, −4.37610349176521967538697203222, −3.37630930377641782693641230227, −2.34536870133876734970716775262, −0.954004844528955802905497898649,
0.48037024905993260663904030957, 2.49999096208060386781603128929, 2.88604818633847601824384843640, 4.08316391921513547361371999350, 4.74233637921707605924347919647, 5.52026051911317394029993809838, 6.24096905844292825732088506983, 7.31143956827787204050314709954, 7.80861615049051351855183274110, 8.593510016125068217966444108090