L(s) = 1 | + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)11-s + (0.173 − 0.984i)13-s + (−0.342 + 0.939i)17-s + (−0.642 + 0.766i)23-s + (−0.342 − 0.939i)29-s + (−0.5 + 0.866i)31-s + 37-s + (−0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (−0.342 − 0.939i)47-s + (0.5 + 0.866i)49-s + (0.766 + 0.642i)53-s + (0.342 − 0.939i)59-s + (0.642 − 0.766i)61-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)11-s + (0.173 − 0.984i)13-s + (−0.342 + 0.939i)17-s + (−0.642 + 0.766i)23-s + (−0.342 − 0.939i)29-s + (−0.5 + 0.866i)31-s + 37-s + (−0.173 − 0.984i)41-s + (−0.766 + 0.642i)43-s + (−0.342 − 0.939i)47-s + (0.5 + 0.866i)49-s + (0.766 + 0.642i)53-s + (0.342 − 0.939i)59-s + (0.642 − 0.766i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1654882627 - 0.6664599410i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1654882627 - 0.6664599410i\) |
\(L(1)\) |
\(\approx\) |
\(0.8586626345 - 0.1663832036i\) |
\(L(1)\) |
\(\approx\) |
\(0.8586626345 - 0.1663832036i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.173 - 0.984i)T \) |
| 17 | \( 1 + (-0.342 + 0.939i)T \) |
| 23 | \( 1 + (-0.642 + 0.766i)T \) |
| 29 | \( 1 + (-0.342 - 0.939i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (-0.342 - 0.939i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + (-0.984 + 0.173i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.342 + 0.939i)T \) |
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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
−18.28549974756332854216718686720, −18.194970905256834450477693096085, −16.97241464403774645585576579845, −16.41071558787790182063434440658, −16.07031886984714682541400585383, −14.9956871737219904762527654297, −14.63583804856390794285760336510, −13.747974911322359532631712124907, −13.126402739317934385101147499480, −12.37750312490839517397141438671, −11.74171366537849144919706144345, −11.2486202369053530726794988916, −10.16474545696280973653147712454, −9.50158622486266385998930994786, −9.1062799553518064316317502828, −8.345661039427497152916360068578, −7.25224151795458771919351326839, −6.71831764673721494110452998865, −6.153306562329177344070036343591, −5.262881784564785732285550868743, −4.3332543629941647822127423788, −3.80663660456634856047646138442, −2.772291764901249172673959606154, −2.13415692879061185173469902165, −1.151663525761161432824690110072,
0.19901732614915894492547810533, 1.20303313564199623938200820661, 2.12318521615891614235451974546, 3.30007645278382100597529385149, 3.630708512741023125382637014061, 4.42696932718516583574508812692, 5.589533265777497145724501251297, 6.09307720693405055396592923484, 6.770844113411725204474607323545, 7.59182386703300299694121118773, 8.32558343107232764593728283014, 9.04656812816668799245987776545, 9.83064263356185047596512304905, 10.377258187514975536484113109193, 11.11228888568199769420353723157, 11.85561210994571432350817044125, 12.63267467606812201498313362755, 13.28724060848500149127312149726, 13.72862181774604058459410407887, 14.669592541620488237775173062204, 15.24089491561641690760656347289, 16.04087942941177543117722374201, 16.549088906280763569271836924116, 17.35100619252984728283507023760, 17.73570019424652266543675030176