L(s) = 1 | + (0.642 − 0.766i)3-s + (0.342 + 0.939i)7-s + (−0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.984 + 0.173i)13-s + (−0.984 + 0.173i)17-s + (−0.766 − 0.642i)19-s + (0.939 + 0.342i)21-s + (−0.866 − 0.5i)23-s + (−0.866 − 0.5i)27-s + (−0.5 − 0.866i)29-s − 31-s + (−0.984 − 0.173i)33-s + (0.766 − 0.642i)39-s + (0.173 − 0.984i)41-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)3-s + (0.342 + 0.939i)7-s + (−0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.984 + 0.173i)13-s + (−0.984 + 0.173i)17-s + (−0.766 − 0.642i)19-s + (0.939 + 0.342i)21-s + (−0.866 − 0.5i)23-s + (−0.866 − 0.5i)27-s + (−0.5 − 0.866i)29-s − 31-s + (−0.984 − 0.173i)33-s + (0.766 − 0.642i)39-s + (0.173 − 0.984i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5536226253 - 1.231611877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5536226253 - 1.231611877i\) |
\(L(1)\) |
\(\approx\) |
\(1.063790981 - 0.4343839764i\) |
\(L(1)\) |
\(\approx\) |
\(1.063790981 - 0.4343839764i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (0.642 - 0.766i)T \) |
| 7 | \( 1 + (0.342 + 0.939i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.984 + 0.173i)T \) |
| 17 | \( 1 + (-0.984 + 0.173i)T \) |
| 19 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.342 - 0.939i)T \) |
| 59 | \( 1 + (0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.342 - 0.939i)T \) |
| 71 | \( 1 + (-0.766 - 0.642i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.984 - 0.173i)T \) |
| 89 | \( 1 + (0.939 + 0.342i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−20.7662537352894289660835635069, −20.19658033443134234422680913057, −19.86318360197340307171468194459, −18.65333498198092606033739821386, −17.93676926415407449377408879329, −17.12293560276255751793147802855, −16.24316770290951891185645199210, −15.72125443592505114516836819451, −14.804139928615577141515386399163, −14.2989254856345629164764724852, −13.31288490386653687460361955739, −12.92787068225884399905257561751, −11.51432700995931379228482190382, −10.75284235484962480476468185773, −10.27618870390086528563166123893, −9.39587262837563957036024431297, −8.54095084007158761006821041444, −7.81317679865015664269704424212, −7.0793417254582041543604676379, −5.90787238858512858973025912418, −4.87831433876003208012350707577, −4.136965641630553575314013835531, −3.56519634336119593277242402595, −2.335953437134691603306182916706, −1.504102248397960985792639205371,
0.4271362960865306202652705261, 1.916949563301199971429463549413, 2.34530317483964543056772245216, 3.43647431385917385763947349458, 4.33354672501910907614763110208, 5.717589920490848802275650867695, 6.14580929338346243935298056611, 7.14689753350742222286830058296, 8.13476504534432880935794955152, 8.7403198741205472011822824468, 9.11051194481771524138015904573, 10.54757901754377541548911128947, 11.28681980822092909192807161549, 12.04220175616172428483575443653, 12.95761097768051414611682353718, 13.47956905206020228688042753902, 14.22033305766747664161532208426, 15.17143168417841062026515195402, 15.62179816356363555157820094202, 16.59818247671429097692684985306, 17.7867199855760594891906923570, 18.13998685591376688726969685093, 18.9770252311101528225851428490, 19.3783150147807861132224464055, 20.4950196855265764853721355819