L(s) = 1 | − 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s − 17-s + ⋯ |
L(s) = 1 | − 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s − 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.009139100024 + 0.2857087708i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.009139100024 + 0.2857087708i\) |
\(L(1)\) |
\(\approx\) |
\(0.3539828748 + 0.2578565877i\) |
\(L(1)\) |
\(\approx\) |
\(0.3539828748 + 0.2578565877i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−28.88156302898669702817944641307, −28.16060031117600827324872089557, −27.07185612254392612485278473156, −26.041434939440388047806520057776, −24.81849026818071786400564386779, −24.02053397203927394167330998561, −23.31673776038384727064096605189, −21.65243375030478554550125066610, −20.172108399936654662580756293233, −19.5239859227038305321522997474, −18.67657341905620044646618038771, −17.29989591035947135304776993393, −16.66794440098146740530087023067, −15.873351116522518228621444444192, −13.88111551139937939177119821906, −12.735428134854227634683818046827, −11.59736762770264704960933690568, −10.78693989045124156512575272901, −9.101207979143254624951280391575, −8.17957382230362473884793439945, −6.96256030418118046954028520402, −6.04296793004238024758516092377, −3.977629153100466067164323879988, −1.810180296080187918053780704768, −0.392744232035325418770258526181,
2.489714583941072471033296079, 3.88527639349470853708360480026, 5.895350136190443929980838982259, 6.813026455512236477357489851244, 8.41406599794053794906199947636, 9.543089251048539351180780694500, 10.52692226084101669178412091360, 11.4253585921731586385016519626, 12.46441397738332229021290923420, 14.98438876994143293892250632343, 15.303647482305156173059495089765, 16.35460723082490699456113111385, 17.65773686978647765736241071261, 18.36543879544339968752929065175, 19.62908693869336478126782021339, 20.52177959876379996236321587784, 21.969203975922329408372079870057, 22.56085570755880606855384211452, 23.91179416516920428560936412963, 25.59753196502485049302271628651, 25.91146010397047704252809040382, 27.32015809201151722089906801535, 27.75984353746352422624538316155, 28.68876595376886999895000510072, 29.80570878103636874292542465349