L(s) = 1 | + (−0.382 + 0.923i)3-s + (−0.980 + 0.195i)5-s + (0.980 + 0.195i)7-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.195 − 0.980i)13-s + (0.195 − 0.980i)15-s + (−0.195 − 0.980i)17-s + (−0.980 + 0.195i)19-s + (−0.555 + 0.831i)21-s + (−0.831 − 0.555i)23-s + (0.923 − 0.382i)25-s + (0.923 − 0.382i)27-s + (0.831 + 0.555i)29-s + (0.923 + 0.382i)31-s + ⋯ |
L(s) = 1 | + (−0.382 + 0.923i)3-s + (−0.980 + 0.195i)5-s + (0.980 + 0.195i)7-s + (−0.707 − 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.195 − 0.980i)13-s + (0.195 − 0.980i)15-s + (−0.195 − 0.980i)17-s + (−0.980 + 0.195i)19-s + (−0.555 + 0.831i)21-s + (−0.831 − 0.555i)23-s + (0.923 − 0.382i)25-s + (0.923 − 0.382i)27-s + (0.831 + 0.555i)29-s + (0.923 + 0.382i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8211889295 - 0.2171628774i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8211889295 - 0.2171628774i\) |
\(L(1)\) |
\(\approx\) |
\(0.8208173584 + 0.06176949671i\) |
\(L(1)\) |
\(\approx\) |
\(0.8208173584 + 0.06176949671i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 97 | \( 1 \) |
good | 3 | \( 1 + (-0.382 + 0.923i)T \) |
| 5 | \( 1 + (-0.980 + 0.195i)T \) |
| 7 | \( 1 + (0.980 + 0.195i)T \) |
| 11 | \( 1 + (0.382 - 0.923i)T \) |
| 13 | \( 1 + (-0.195 - 0.980i)T \) |
| 17 | \( 1 + (-0.195 - 0.980i)T \) |
| 19 | \( 1 + (-0.980 + 0.195i)T \) |
| 23 | \( 1 + (-0.831 - 0.555i)T \) |
| 29 | \( 1 + (0.831 + 0.555i)T \) |
| 31 | \( 1 + (0.923 + 0.382i)T \) |
| 37 | \( 1 + (-0.555 - 0.831i)T \) |
| 41 | \( 1 + (0.555 - 0.831i)T \) |
| 43 | \( 1 + (-0.707 + 0.707i)T \) |
| 47 | \( 1 + (0.707 - 0.707i)T \) |
| 53 | \( 1 + (-0.382 - 0.923i)T \) |
| 59 | \( 1 + (0.831 - 0.555i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.195 + 0.980i)T \) |
| 71 | \( 1 + (-0.555 - 0.831i)T \) |
| 73 | \( 1 + (0.707 + 0.707i)T \) |
| 79 | \( 1 + (-0.923 - 0.382i)T \) |
| 83 | \( 1 + (-0.980 + 0.195i)T \) |
| 89 | \( 1 + (-0.382 + 0.923i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.22025113433050938379200668302, −23.76786292836301157547098708793, −23.159207432111164848208373228835, −22.08206398571870794503552099773, −21.01664475361618769159823810074, −19.91020306573473641911719436906, −19.36780075865844389793724766417, −18.48031380405161932645001142388, −17.31090510200693432783057899037, −17.06565041361049758789065192689, −15.64649880303449597255708155911, −14.74908059441816306612151333879, −13.87175352751931015336301303090, −12.72071390973174935313076548544, −11.88811998093330812016904962101, −11.42663162717728548894359452963, −10.28259363835736221214574618052, −8.68029675229709982581296319655, −7.99677275942038810334386907326, −7.12599532782012996359250052339, −6.2363050034998910481526538816, −4.72482438669086390438125363829, −4.13601029958670398719181983097, −2.249420589154396594931453384638, −1.30703539925799943937424316421,
0.59030446005716250753888478425, 2.696093758577661184905994831671, 3.79278956251586624587072029815, 4.680849523503757052164426067697, 5.598988763700799589648731347985, 6.83689180108377153812221323985, 8.25297341349770349210437210256, 8.64976203207003080487477901843, 10.17134684770910241826077324316, 10.94281631963321420925236542784, 11.63999936706455251359934636735, 12.42303805248691087984043519493, 14.117222801984136544659525640373, 14.72111484391379555094360474629, 15.68801310875297614904870397018, 16.23702940725377278377654185843, 17.3667405769797296756111826883, 18.12602008466652928799019925893, 19.27118341686958786807576334758, 20.18699939830756171578750569161, 20.959770484651707730044983538518, 21.86141850161603898796156594399, 22.630051563010356677499097403550, 23.41743896617408817135557334142, 24.301331002381972815355174792875