L(s) = 1 | + (0.473 + 0.880i)2-s + (−0.0448 − 0.998i)3-s + (−0.550 + 0.834i)4-s + (0.309 − 0.951i)5-s + (0.858 − 0.512i)6-s + (−0.983 − 0.178i)7-s + (−0.995 − 0.0896i)8-s + (−0.995 + 0.0896i)9-s + (0.983 − 0.178i)10-s + (−0.753 + 0.657i)11-s + (0.858 + 0.512i)12-s + (−0.753 − 0.657i)13-s + (−0.309 − 0.951i)14-s + (−0.963 − 0.266i)15-s + (−0.393 − 0.919i)16-s + (0.809 − 0.587i)17-s + ⋯ |
L(s) = 1 | + (0.473 + 0.880i)2-s + (−0.0448 − 0.998i)3-s + (−0.550 + 0.834i)4-s + (0.309 − 0.951i)5-s + (0.858 − 0.512i)6-s + (−0.983 − 0.178i)7-s + (−0.995 − 0.0896i)8-s + (−0.995 + 0.0896i)9-s + (0.983 − 0.178i)10-s + (−0.753 + 0.657i)11-s + (0.858 + 0.512i)12-s + (−0.753 − 0.657i)13-s + (−0.309 − 0.951i)14-s + (−0.963 − 0.266i)15-s + (−0.393 − 0.919i)16-s + (0.809 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3907781487 - 0.6754154948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3907781487 - 0.6754154948i\) |
\(L(1)\) |
\(\approx\) |
\(0.8853921287 - 0.1108345086i\) |
\(L(1)\) |
\(\approx\) |
\(0.8853921287 - 0.1108345086i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 \) |
good | 2 | \( 1 + (0.473 + 0.880i)T \) |
| 3 | \( 1 + (-0.0448 - 0.998i)T \) |
| 5 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (-0.983 - 0.178i)T \) |
| 11 | \( 1 + (-0.753 + 0.657i)T \) |
| 13 | \( 1 + (-0.753 - 0.657i)T \) |
| 17 | \( 1 + (0.809 - 0.587i)T \) |
| 19 | \( 1 + (-0.963 + 0.266i)T \) |
| 23 | \( 1 + (0.900 - 0.433i)T \) |
| 29 | \( 1 + (0.936 - 0.351i)T \) |
| 31 | \( 1 + (0.393 - 0.919i)T \) |
| 37 | \( 1 + (-0.900 - 0.433i)T \) |
| 41 | \( 1 + (0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.691 + 0.722i)T \) |
| 47 | \( 1 + (0.0448 - 0.998i)T \) |
| 53 | \( 1 + (0.550 + 0.834i)T \) |
| 59 | \( 1 + (-0.858 - 0.512i)T \) |
| 61 | \( 1 + (-0.983 + 0.178i)T \) |
| 67 | \( 1 + (0.550 - 0.834i)T \) |
| 73 | \( 1 + (0.473 + 0.880i)T \) |
| 79 | \( 1 + (-0.995 - 0.0896i)T \) |
| 83 | \( 1 + (0.858 + 0.512i)T \) |
| 89 | \( 1 + (-0.550 - 0.834i)T \) |
| 97 | \( 1 + (0.222 - 0.974i)T \) |
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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
−31.88478472991462483266302700064, −30.79826678525312277864247616612, −29.33833262358785755757695026661, −28.90912341346249629922402608592, −27.50455991393795372476258861513, −26.53296613551612629404713449437, −25.60919106056599228345248648375, −23.56978783725720757643757684280, −22.68668444139539067752017495040, −21.62180299067541676718330097317, −21.26241925745657838170031301557, −19.503979007162919071294994599277, −18.897886424687589607938734174544, −17.24150198014658587836364717403, −15.704282972202252679374014802766, −14.68461762986666157995749595534, −13.62995891524664019757998000382, −12.15234219921439427333459593592, −10.72335019061080227078299593450, −10.14956984576287604154405832018, −8.94436238136587559383488196904, −6.43701364718631279273678909623, −5.19414494264419720701697672938, −3.53426701697204421567184642164, −2.65801386164465939372827684610,
0.31875466150606982453300613432, 2.767540905710711288152705789341, 4.8539799570435170165434617412, 6.02374032607937946239585210728, 7.27601288636207700441917901999, 8.35328294812501694570829539193, 9.79870955828337879183327730811, 12.32113374706945915390697401397, 12.78468737002705851467739472247, 13.72923352693698148431232683734, 15.20528749953342781994419765926, 16.601897591295355558005021192055, 17.30321015972925553290109694895, 18.55793268125365599052115977377, 19.95363604083545468734905634295, 21.21050348201800019032409839998, 22.878191852745495943953282802287, 23.322528726956587698696477675005, 24.72050079466150648026578527569, 25.203813007327443586976771735414, 26.23410262116037309754863179808, 27.89264273255024648739806515131, 29.15236757222874011696957465682, 29.98324627881309999812098099507, 31.49045544632563583668454502055