L(s) = 1 | − 5-s − 1.41i·7-s + 11-s − 13-s − 17-s + 19-s − 23-s − 1.41i·31-s + 1.41i·35-s − 1.41i·37-s − 41-s − 43-s − 1.41i·47-s − 1.00·49-s + 1.41i·53-s + ⋯ |
L(s) = 1 | − 5-s − 1.41i·7-s + 11-s − 13-s − 17-s + 19-s − 23-s − 1.41i·31-s + 1.41i·35-s − 1.41i·37-s − 41-s − 43-s − 1.41i·47-s − 1.00·49-s + 1.41i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6358834322\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6358834322\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 1.41iT - T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.849846826846949466904030159627, −7.86879314197000157011998814481, −7.34676028028109217740493551294, −6.85432112542552520359496177537, −5.78369191088288333643128117643, −4.51789993306017833989556489420, −4.11710110675862527828872684700, −3.36437854922545179748816957716, −1.91616987306121324036095325205, −0.41591831571272995581069633124,
1.72376226840923333116863171027, 2.82894983947271685607974807952, 3.69633514139184534115929782871, 4.71106247976782240738257836779, 5.35158155491158432899602951978, 6.47296484370388385668093279473, 6.98176547860601372823561444636, 8.130300908233192138743690179919, 8.443662066419726348029924415745, 9.430764566231257938017958051063