| L(s) = 1 | + 1.10·2-s − 1.62·3-s − 0.788·4-s + 2.17·5-s − 1.78·6-s + 4.08·7-s − 3.06·8-s − 0.369·9-s + 2.39·10-s + 3.06·11-s + 1.27·12-s − 1.48·13-s + 4.49·14-s − 3.53·15-s − 1.80·16-s − 17-s − 0.406·18-s + 2.86·19-s − 1.71·20-s − 6.62·21-s + 3.37·22-s + 0.388·23-s + 4.97·24-s − 0.249·25-s − 1.62·26-s + 5.46·27-s − 3.22·28-s + ⋯ |
| L(s) = 1 | + 0.778·2-s − 0.936·3-s − 0.394·4-s + 0.974·5-s − 0.729·6-s + 1.54·7-s − 1.08·8-s − 0.123·9-s + 0.758·10-s + 0.923·11-s + 0.368·12-s − 0.410·13-s + 1.20·14-s − 0.912·15-s − 0.450·16-s − 0.242·17-s − 0.0957·18-s + 0.657·19-s − 0.384·20-s − 1.44·21-s + 0.718·22-s + 0.0809·23-s + 1.01·24-s − 0.0498·25-s − 0.319·26-s + 1.05·27-s − 0.608·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.903822678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.903822678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 3 | \( 1 + 1.62T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 - 3.06T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 23 | \( 1 - 0.388T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 - 8.20T + 37T^{2} \) |
| 41 | \( 1 - 8.12T + 41T^{2} \) |
| 47 | \( 1 + 0.818T + 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 9.50T + 67T^{2} \) |
| 71 | \( 1 + 5.02T + 71T^{2} \) |
| 73 | \( 1 - 0.515T + 73T^{2} \) |
| 79 | \( 1 + 7.81T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 1.59T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.55345030289377776431942923404, −9.554029241357021523609622261796, −8.793487809370030975846617887674, −7.76295607676559296531630226118, −6.29716097772755420151765759944, −5.88057903661326217421411518000, −4.83347235221004013104651798530, −4.49904613296870697537831344283, −2.75892389012738805974062451428, −1.20445355868727055390263502011,
1.20445355868727055390263502011, 2.75892389012738805974062451428, 4.49904613296870697537831344283, 4.83347235221004013104651798530, 5.88057903661326217421411518000, 6.29716097772755420151765759944, 7.76295607676559296531630226118, 8.793487809370030975846617887674, 9.554029241357021523609622261796, 10.55345030289377776431942923404
