L(s) = 1 | + (−0.602 − 0.798i)2-s + (0.445 + 0.895i)3-s + (−0.273 + 0.961i)4-s + (−0.273 − 0.961i)5-s + (0.445 − 0.895i)6-s + (0.739 − 0.673i)7-s + (0.932 − 0.361i)8-s + (−0.602 + 0.798i)9-s + (−0.602 + 0.798i)10-s + (0.445 − 0.895i)11-s + (−0.982 + 0.183i)12-s + (−0.850 − 0.526i)13-s + (−0.982 − 0.183i)14-s + (0.739 − 0.673i)15-s + (−0.850 − 0.526i)16-s + 17-s + ⋯ |
L(s) = 1 | + (−0.602 − 0.798i)2-s + (0.445 + 0.895i)3-s + (−0.273 + 0.961i)4-s + (−0.273 − 0.961i)5-s + (0.445 − 0.895i)6-s + (0.739 − 0.673i)7-s + (0.932 − 0.361i)8-s + (−0.602 + 0.798i)9-s + (−0.602 + 0.798i)10-s + (0.445 − 0.895i)11-s + (−0.982 + 0.183i)12-s + (−0.850 − 0.526i)13-s + (−0.982 − 0.183i)14-s + (0.739 − 0.673i)15-s + (−0.850 − 0.526i)16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6468810410 - 0.6830207989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6468810410 - 0.6830207989i\) |
\(L(1)\) |
\(\approx\) |
\(0.7993616729 - 0.3403705201i\) |
\(L(1)\) |
\(\approx\) |
\(0.7993616729 - 0.3403705201i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 307 | \( 1 \) |
good | 2 | \( 1 + (-0.602 - 0.798i)T \) |
| 3 | \( 1 + (0.445 + 0.895i)T \) |
| 5 | \( 1 + (-0.273 - 0.961i)T \) |
| 7 | \( 1 + (0.739 - 0.673i)T \) |
| 11 | \( 1 + (0.445 - 0.895i)T \) |
| 13 | \( 1 + (-0.850 - 0.526i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.850 - 0.526i)T \) |
| 23 | \( 1 + (-0.602 + 0.798i)T \) |
| 29 | \( 1 + (-0.602 - 0.798i)T \) |
| 31 | \( 1 + (0.445 - 0.895i)T \) |
| 37 | \( 1 + (0.739 - 0.673i)T \) |
| 41 | \( 1 + (0.739 + 0.673i)T \) |
| 43 | \( 1 + (0.932 - 0.361i)T \) |
| 47 | \( 1 + (-0.850 - 0.526i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.0922 - 0.995i)T \) |
| 61 | \( 1 + (0.932 - 0.361i)T \) |
| 67 | \( 1 + (-0.982 + 0.183i)T \) |
| 71 | \( 1 + (0.932 + 0.361i)T \) |
| 73 | \( 1 + (-0.850 - 0.526i)T \) |
| 79 | \( 1 + (0.932 + 0.361i)T \) |
| 83 | \( 1 + (0.739 + 0.673i)T \) |
| 89 | \( 1 + (-0.982 + 0.183i)T \) |
| 97 | \( 1 + (-0.850 + 0.526i)T \) |
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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
−25.558811073817903706344734302995, −24.74120929621905862598052164400, −23.95276292793684697578902567288, −23.13666600606351821325333657295, −22.2613507329014275221699880720, −20.86219406734733310913628208135, −19.61478078521979247685254561044, −19.0559811561901380278577579659, −18.21743289330725538777374868737, −17.67812172207705243715832967404, −16.58781616979892530087002144903, −15.12205052946633689451884502447, −14.556514570264038074636232314304, −14.22468556368501827836173050819, −12.50700385999566310370760945449, −11.71100600910469691733465797568, −10.417701783102105450249663833341, −9.35117039006603904839917519416, −8.28445481448443748324870689085, −7.51140130917739914105242503958, −6.76158363283146234344755342792, −5.797475803647989735841067411952, −4.33280395819932736808769767808, −2.52707690056243161375230649155, −1.60615632495339174946499956179,
0.74032282725203399044678517205, 2.25129535931112816232038964072, 3.6818309496624504896280745578, 4.33041141697624472581979637899, 5.44222438799259663303384431766, 7.74830601573594469960679994901, 8.14178423046723870375883356785, 9.250461808550199533947016700486, 9.97106055872959844324311853734, 11.07353162792265846170485396100, 11.75100520044044166041480779514, 13.02280789035764032761565783455, 13.919179671410132789759379102607, 15.02812034148968106048307341645, 16.31832640179981712012818855691, 16.89970560237061293151435245387, 17.58610661896190249794954214647, 19.25546799190585662485260036855, 19.69224109902231878835335071740, 20.60109454916574001177056885336, 21.19740663150723197942552352100, 21.92142896919284129700482039973, 23.118517687387177771190845667284, 24.33483633501771991991189992393, 25.20774543053787086902997389563