| L(s) = 1 | − 2-s − 0.600·3-s + 4-s + 0.151·5-s + 0.600·6-s − 2.68·7-s − 8-s − 2.63·9-s − 0.151·10-s − 3.19·11-s − 0.600·12-s − 3.37·13-s + 2.68·14-s − 0.0909·15-s + 16-s − 5.05·17-s + 2.63·18-s − 3.19·19-s + 0.151·20-s + 1.61·21-s + 3.19·22-s − 4.99·23-s + 0.600·24-s − 4.97·25-s + 3.37·26-s + 3.38·27-s − 2.68·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.346·3-s + 0.5·4-s + 0.0677·5-s + 0.245·6-s − 1.01·7-s − 0.353·8-s − 0.879·9-s − 0.0478·10-s − 0.963·11-s − 0.173·12-s − 0.937·13-s + 0.718·14-s − 0.0234·15-s + 0.250·16-s − 1.22·17-s + 0.622·18-s − 0.733·19-s + 0.0338·20-s + 0.352·21-s + 0.681·22-s − 1.04·23-s + 0.122·24-s − 0.995·25-s + 0.662·26-s + 0.651·27-s − 0.507·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07889830376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07889830376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 0.600T + 3T^{2} \) |
| 5 | \( 1 - 0.151T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 3.19T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 + 1.94T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 3.13T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 + 9.20T + 47T^{2} \) |
| 53 | \( 1 + 7.20T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 + 3.02T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 4.86T + 73T^{2} \) |
| 79 | \( 1 + 9.72T + 79T^{2} \) |
| 83 | \( 1 - 3.42T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 0.970T + 97T^{2} \) |
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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
−8.267124689864847629164478727278, −7.980798747626964435717354098335, −6.85898063175991180838347881161, −6.37678356729635951432133199249, −5.65936292280705584447436989076, −4.81432950312168107274896633899, −3.70757997416003544850393839009, −2.65191497279381722941986416147, −2.12033799806854950999485333517, −0.16459671809721121831537738326,
0.16459671809721121831537738326, 2.12033799806854950999485333517, 2.65191497279381722941986416147, 3.70757997416003544850393839009, 4.81432950312168107274896633899, 5.65936292280705584447436989076, 6.37678356729635951432133199249, 6.85898063175991180838347881161, 7.980798747626964435717354098335, 8.267124689864847629164478727278
