| L(s) = 1 | + 0.381·2-s − 2·3-s − 1.85·4-s − 2.23·5-s − 0.763·6-s + 7-s − 1.47·8-s + 9-s − 0.854·10-s + 4.47·11-s + 3.70·12-s + 13-s + 0.381·14-s + 4.47·15-s + 3.14·16-s + 5.23·17-s + 0.381·18-s + 19-s + 4.14·20-s − 2·21-s + 1.70·22-s − 5.23·23-s + 2.94·24-s + 0.381·26-s + 4·27-s − 1.85·28-s + 0.763·29-s + ⋯ |
| L(s) = 1 | + 0.270·2-s − 1.15·3-s − 0.927·4-s − 0.999·5-s − 0.311·6-s + 0.377·7-s − 0.520·8-s + 0.333·9-s − 0.270·10-s + 1.34·11-s + 1.07·12-s + 0.277·13-s + 0.102·14-s + 1.15·15-s + 0.786·16-s + 1.26·17-s + 0.0900·18-s + 0.229·19-s + 0.927·20-s − 0.436·21-s + 0.364·22-s − 1.09·23-s + 0.600·24-s + 0.0749·26-s + 0.769·27-s − 0.350·28-s + 0.141·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7178322948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7178322948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 - 0.763T + 29T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 - 0.291T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 + 0.763T + 89T^{2} \) |
| 97 | \( 1 + 4.70T + 97T^{2} \) |
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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
−11.71059178098979551381226891670, −10.51592952865741377720638019614, −9.486181981244515634186336117324, −8.458214002869180016692885907160, −7.57332373861825827803598512192, −6.22988974312289118217788932561, −5.45333210532548463527821809569, −4.34226250947803407534919578169, −3.62830689599313504451930849704, −0.856550167488161402501682890800,
0.856550167488161402501682890800, 3.62830689599313504451930849704, 4.34226250947803407534919578169, 5.45333210532548463527821809569, 6.22988974312289118217788932561, 7.57332373861825827803598512192, 8.458214002869180016692885907160, 9.486181981244515634186336117324, 10.51592952865741377720638019614, 11.71059178098979551381226891670
