L(s) = 1 | + (−0.959 + 0.281i)7-s + (0.415 − 0.909i)11-s + (0.654 + 0.755i)13-s + (0.841 + 0.540i)17-s + (0.959 + 0.281i)19-s + (0.142 + 0.989i)23-s − 29-s + (0.654 − 0.755i)31-s − 37-s + (−0.841 − 0.540i)41-s + (0.841 + 0.540i)43-s + (0.142 + 0.989i)47-s + (0.841 − 0.540i)49-s + (0.841 − 0.540i)53-s + (−0.654 + 0.755i)59-s + ⋯ |
L(s) = 1 | + (−0.959 + 0.281i)7-s + (0.415 − 0.909i)11-s + (0.654 + 0.755i)13-s + (0.841 + 0.540i)17-s + (0.959 + 0.281i)19-s + (0.142 + 0.989i)23-s − 29-s + (0.654 − 0.755i)31-s − 37-s + (−0.841 − 0.540i)41-s + (0.841 + 0.540i)43-s + (0.142 + 0.989i)47-s + (0.841 − 0.540i)49-s + (0.841 − 0.540i)53-s + (−0.654 + 0.755i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.387851800 + 0.7371293443i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.387851800 + 0.7371293443i\) |
\(L(1)\) |
\(\approx\) |
\(1.046274458 + 0.1283899169i\) |
\(L(1)\) |
\(\approx\) |
\(1.046274458 + 0.1283899169i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 \) |
good | 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 11 | \( 1 + (0.415 - 0.909i)T \) |
| 13 | \( 1 + (0.654 + 0.755i)T \) |
| 17 | \( 1 + (0.841 + 0.540i)T \) |
| 19 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.142 + 0.989i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.654 - 0.755i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.841 - 0.540i)T \) |
| 43 | \( 1 + (0.841 + 0.540i)T \) |
| 47 | \( 1 + (0.142 + 0.989i)T \) |
| 53 | \( 1 + (0.841 - 0.540i)T \) |
| 59 | \( 1 + (-0.654 + 0.755i)T \) |
| 61 | \( 1 + (0.415 + 0.909i)T \) |
| 71 | \( 1 + (0.841 - 0.540i)T \) |
| 73 | \( 1 + (-0.415 - 0.909i)T \) |
| 79 | \( 1 + (0.654 + 0.755i)T \) |
| 83 | \( 1 + (-0.415 + 0.909i)T \) |
| 89 | \( 1 + (0.142 - 0.989i)T \) |
| 97 | \( 1 - T \) |
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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
−18.49615579498918929751159388270, −17.65154262162702915981557411068, −16.99460783735246497360415395391, −16.30932625895576583569753749577, −15.674336922670863643304016848086, −15.09838504731307855778154759240, −14.16176318496708245626437270299, −13.641747112662192780005021445408, −12.80623126374445376897256079948, −12.30820648914581664285861776341, −11.610496247349328185184779568495, −10.59140790296691832224147251748, −10.091043698500393896154318732, −9.43779472501723460955645106462, −8.727895598050960858959467821328, −7.79709413775917413910843397977, −7.059853417774514920600368480203, −6.56720830714521802523391642710, −5.5839306918278230963890058045, −4.981502079209416595614807056572, −3.90823292871422123000470653076, −3.328918430277604571282842966655, −2.57635906227646514914968348995, −1.42749147459843627540941288038, −0.5503928767408337577780777050,
0.927673377980015775269483103923, 1.7172565312832134448445845126, 2.89182828435596269965453938581, 3.56579227449297815034956775810, 4.02823785903330558267647596822, 5.42514130158056164531423111323, 5.83144736306714482383220275043, 6.54521640854017348649232785191, 7.35880737404305372584667016537, 8.15625530757431363559402436701, 9.03239294852916052429684922149, 9.448531440654940834459402664065, 10.2332060054955251851647893155, 11.08889865766770472456794228283, 11.777290001606603547572157928020, 12.301880464542288674663241042840, 13.30244029640732526132552536902, 13.692415449062070751736364444139, 14.4018407769425124201239575363, 15.33185073627072077743005374402, 15.90052237566664391260507121682, 16.60470425984265431213251952364, 16.964811044764535006726729371557, 18.0131095450369542110003810369, 18.77098519915282599086505544184