| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.514193680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.514193680\) |
| \(L(1)\) |
\(\approx\) |
\(1.034503778\) |
| \(L(1)\) |
\(\approx\) |
\(1.034503778\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 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
−30.42625828885605464404754053425, −29.8175414743446227156649901676, −28.00655730668914839263179214377, −27.19646908631341211815177959955, −26.763907261751893345420094554507, −25.27120794014394381586801708542, −24.60413059275266014089825643164, −23.52164458554992604146030289458, −21.52541122809908710073631738364, −20.574298407733945331186303160384, −19.482550901856165576555907252030, −19.0508802564173414066333622989, −17.563391217363846108557770649288, −16.41001684935351027898122473984, −14.963328146137910103109816042148, −14.59568381529952022151224491492, −12.409656206446733262351632968997, −11.39863095518579283183326661167, −9.9944528157185826666408088977, −8.68516345573448593133850598003, −7.92691097611334965494046296489, −6.91759279924790059190355476608, −4.43836820028830768840739500457, −2.85164903161242844229259076713, −1.22292048422529503187794249765,
1.22292048422529503187794249765, 2.85164903161242844229259076713, 4.43836820028830768840739500457, 6.91759279924790059190355476608, 7.92691097611334965494046296489, 8.68516345573448593133850598003, 9.9944528157185826666408088977, 11.39863095518579283183326661167, 12.409656206446733262351632968997, 14.59568381529952022151224491492, 14.963328146137910103109816042148, 16.41001684935351027898122473984, 17.563391217363846108557770649288, 19.0508802564173414066333622989, 19.482550901856165576555907252030, 20.574298407733945331186303160384, 21.52541122809908710073631738364, 23.52164458554992604146030289458, 24.60413059275266014089825643164, 25.27120794014394381586801708542, 26.763907261751893345420094554507, 27.19646908631341211815177959955, 28.00655730668914839263179214377, 29.8175414743446227156649901676, 30.42625828885605464404754053425
