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
−8.593510016125068217966444108090, −7.80861615049051351855183274110, −7.31143956827787204050314709954, −6.24096905844292825732088506983, −5.52026051911317394029993809838, −4.74233637921707605924347919647, −4.08316391921513547361371999350, −2.88604818633847601824384843640, −2.49999096208060386781603128929, −0.48037024905993260663904030957,
0.954004844528955802905497898649, 2.34536870133876734970716775262, 3.37630930377641782693641230227, 4.37610349176521967538697203222, 4.73338849065058476795574324906, 5.77100984853167728117966741393, 6.44151499162125857060873929320, 7.38015917477939998914327831035, 7.84414948927733129577888525763, 8.897746560270578670220301141329