| L(s) = 1 | + 5-s + 7-s − 11-s − 17-s + 19-s + 23-s + 25-s − 29-s + 31-s + 35-s − 37-s + 41-s − 43-s − 47-s + 49-s − 53-s − 55-s − 59-s + 61-s + 67-s − 71-s − 73-s − 77-s − 79-s − 83-s − 85-s + 89-s + ⋯ |
| L(s) = 1 | + 5-s + 7-s − 11-s − 17-s + 19-s + 23-s + 25-s − 29-s + 31-s + 35-s − 37-s + 41-s − 43-s − 47-s + 49-s − 53-s − 55-s − 59-s + 61-s + 67-s − 71-s − 73-s − 77-s − 79-s − 83-s − 85-s + 89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.354834269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.354834269\) |
| \(L(1)\) |
\(\approx\) |
\(1.252721863\) |
| \(L(1)\) |
\(\approx\) |
\(1.252721863\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 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 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 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
−28.08952342959007179161160715902, −26.7579643004131428186736903311, −26.12394924533076454881977002204, −24.7873657472678373018236030165, −24.31284846683850010093004092883, −23.01672915020626418507576496948, −21.902513414580666518237172200155, −21.000219605883609297952142847191, −20.38155456210698251274803821420, −18.78686852637687041741577050955, −17.88750083597847765572759541528, −17.248170998060921577746507807268, −15.88252518940245274182278967484, −14.781253505961444793689414707315, −13.73443711993739959590391021541, −12.947336067023999121031859885134, −11.44175102311064761402949027364, −10.53556325109306371139124839751, −9.368769654136979694025704936688, −8.22894322212505799813373706391, −6.98801470255497230193115773101, −5.5557364183684641992903074449, −4.74644557251703587719248327456, −2.8170057619306344697984671392, −1.58858253531340228208737911886,
1.58858253531340228208737911886, 2.8170057619306344697984671392, 4.74644557251703587719248327456, 5.5557364183684641992903074449, 6.98801470255497230193115773101, 8.22894322212505799813373706391, 9.368769654136979694025704936688, 10.53556325109306371139124839751, 11.44175102311064761402949027364, 12.947336067023999121031859885134, 13.73443711993739959590391021541, 14.781253505961444793689414707315, 15.88252518940245274182278967484, 17.248170998060921577746507807268, 17.88750083597847765572759541528, 18.78686852637687041741577050955, 20.38155456210698251274803821420, 21.000219605883609297952142847191, 21.902513414580666518237172200155, 23.01672915020626418507576496948, 24.31284846683850010093004092883, 24.7873657472678373018236030165, 26.12394924533076454881977002204, 26.7579643004131428186736903311, 28.08952342959007179161160715902
