L(s) = 1 | + (−0.754 + 0.656i)2-s + (−0.889 + 0.456i)3-s + (0.137 − 0.990i)4-s + (0.509 + 0.860i)5-s + (0.371 − 0.928i)6-s + (0.894 + 0.446i)7-s + (0.546 + 0.837i)8-s + (0.583 − 0.812i)9-s + (−0.949 − 0.314i)10-s + (0.202 − 0.979i)11-s + (0.329 + 0.944i)12-s + (−0.942 + 0.335i)13-s + (−0.968 + 0.250i)14-s + (−0.846 − 0.533i)15-s + (−0.962 − 0.272i)16-s + (−0.556 + 0.831i)17-s + ⋯ |
L(s) = 1 | + (−0.754 + 0.656i)2-s + (−0.889 + 0.456i)3-s + (0.137 − 0.990i)4-s + (0.509 + 0.860i)5-s + (0.371 − 0.928i)6-s + (0.894 + 0.446i)7-s + (0.546 + 0.837i)8-s + (0.583 − 0.812i)9-s + (−0.949 − 0.314i)10-s + (0.202 − 0.979i)11-s + (0.329 + 0.944i)12-s + (−0.942 + 0.335i)13-s + (−0.968 + 0.250i)14-s + (−0.846 − 0.533i)15-s + (−0.962 − 0.272i)16-s + (−0.556 + 0.831i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2745800267 + 0.7066365401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2745800267 + 0.7066365401i\) |
\(L(1)\) |
\(\approx\) |
\(0.5422571915 + 0.4025421226i\) |
\(L(1)\) |
\(\approx\) |
\(0.5422571915 + 0.4025421226i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (-0.754 + 0.656i)T \) |
| 3 | \( 1 + (-0.889 + 0.456i)T \) |
| 5 | \( 1 + (0.509 + 0.860i)T \) |
| 7 | \( 1 + (0.894 + 0.446i)T \) |
| 11 | \( 1 + (0.202 - 0.979i)T \) |
| 13 | \( 1 + (-0.942 + 0.335i)T \) |
| 17 | \( 1 + (-0.556 + 0.831i)T \) |
| 19 | \( 1 + (0.988 + 0.153i)T \) |
| 23 | \( 1 + (-0.934 + 0.355i)T \) |
| 29 | \( 1 + (0.851 - 0.523i)T \) |
| 31 | \( 1 + (0.245 + 0.969i)T \) |
| 37 | \( 1 + (0.959 - 0.282i)T \) |
| 41 | \( 1 + (0.975 - 0.218i)T \) |
| 43 | \( 1 + (0.00551 - 0.999i)T \) |
| 47 | \( 1 + (0.716 - 0.697i)T \) |
| 53 | \( 1 + (-0.857 + 0.514i)T \) |
| 59 | \( 1 + (-0.986 + 0.164i)T \) |
| 61 | \( 1 + (-0.421 + 0.906i)T \) |
| 67 | \( 1 + (0.874 + 0.485i)T \) |
| 71 | \( 1 + (-0.104 + 0.994i)T \) |
| 73 | \( 1 + (-0.234 + 0.972i)T \) |
| 79 | \( 1 + (0.528 + 0.849i)T \) |
| 83 | \( 1 + (-0.989 + 0.142i)T \) |
| 89 | \( 1 + (0.618 + 0.785i)T \) |
| 97 | \( 1 + (0.984 - 0.175i)T \) |
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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
−22.70441877979979769331527876082, −22.053968121184216784304152675390, −21.158748061412292247820294031694, −20.14699769904073135536342416219, −19.9446778523013749878178509406, −18.42936013088211987268710187664, −17.74229705949335446688194073893, −17.44572884666391404940435638192, −16.588347843450247304921137172063, −15.7836341231008246771549175591, −14.19285127306124152140587325754, −13.25686279956208383901076180329, −12.382786713215505747367080477235, −11.85536882907559278227608114176, −10.96286138907669837767932799687, −9.9350162231534836569551145513, −9.378509933534207281210098099, −7.91201519879171874758875604880, −7.51058834036300544018324398315, −6.29904293287618194217808617595, −4.76504393656385552787100314129, −4.58106790372539718524484882402, −2.478002343379954466873774485865, −1.60929516596011587285741716465, −0.64241145581743564738136257370,
1.26467257892685654436490558104, 2.488531385709067508519566809284, 4.18806943284072243472102916989, 5.39471427065346487121037154657, 5.92851716030751895598358669309, 6.81791796842506769746455018082, 7.80793371713462029043170015154, 8.935403320849009883221022374821, 9.82084807532162381151745718556, 10.60954010058713836374974029601, 11.304959288158302742958572270200, 12.07795863227186259830645171636, 13.860949352591239506949087496190, 14.43111938565882485165720454731, 15.34134796342163583924435829510, 16.03098983668118027547796320728, 17.12777196126937141770668627913, 17.583795184473645820594594466877, 18.29065279492824429410021494770, 19.03392727726235491638042666688, 20.096877665662162594826630688568, 21.574880387785149825597473410520, 21.71617790230770739169448321017, 22.76121595801551689579887319842, 23.78162377271153668962563327542