| L(s) = 1 | + 1.52·2-s − 5.67·4-s − 9.65·5-s + 22.3·7-s − 20.8·8-s − 14.7·10-s + 50.3·11-s + 34.0·14-s + 13.6·16-s − 86.1·17-s + 116.·19-s + 54.8·20-s + 76.6·22-s − 72·23-s − 31.7·25-s − 126.·28-s − 14.1·29-s − 196.·31-s + 187.·32-s − 131.·34-s − 216·35-s − 154.·37-s + 178.·38-s + 201.·40-s + 265.·41-s + 211.·43-s − 285.·44-s + ⋯ |
| L(s) = 1 | + 0.538·2-s − 0.709·4-s − 0.863·5-s + 1.20·7-s − 0.921·8-s − 0.465·10-s + 1.37·11-s + 0.650·14-s + 0.213·16-s − 1.22·17-s + 1.41·19-s + 0.613·20-s + 0.742·22-s − 0.652·23-s − 0.253·25-s − 0.857·28-s − 0.0905·29-s − 1.13·31-s + 1.03·32-s − 0.662·34-s − 1.04·35-s − 0.686·37-s + 0.760·38-s + 0.795·40-s + 1.01·41-s + 0.751·43-s − 0.978·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.064939756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064939756\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.52T + 8T^{2} \) |
| 5 | \( 1 + 9.65T + 125T^{2} \) |
| 7 | \( 1 - 22.3T + 343T^{2} \) |
| 11 | \( 1 - 50.3T + 1.33e3T^{2} \) |
| 17 | \( 1 + 86.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 + 14.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 265.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 686.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 91.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 329.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 768.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 771.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 514.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 527.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 74.2T + 9.12e5T^{2} \) |
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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
−9.111704209069981317498417220622, −8.258229188059038619332444661586, −7.63247667352890777744779949403, −6.63300568408903600366030158176, −5.59157550818188078345752440370, −4.75565083510645587656220721721, −4.07781631222911387694578150741, −3.47099208295994480516293997481, −1.88749426711453453609746327506, −0.66467533455827391312761878823,
0.66467533455827391312761878823, 1.88749426711453453609746327506, 3.47099208295994480516293997481, 4.07781631222911387694578150741, 4.75565083510645587656220721721, 5.59157550818188078345752440370, 6.63300568408903600366030158176, 7.63247667352890777744779949403, 8.258229188059038619332444661586, 9.111704209069981317498417220622
