| L(s) = 1 | + 2.30·2-s + 3-s + 3.33·4-s − 1.64·5-s + 2.30·6-s − 7-s + 3.07·8-s + 9-s − 3.79·10-s − 0.686·11-s + 3.33·12-s + 6.93·13-s − 2.30·14-s − 1.64·15-s + 0.441·16-s + 2.02·17-s + 2.30·18-s − 0.821·19-s − 5.47·20-s − 21-s − 1.58·22-s + 6.97·23-s + 3.07·24-s − 2.30·25-s + 16.0·26-s + 27-s − 3.33·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 0.577·3-s + 1.66·4-s − 0.734·5-s + 0.942·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s − 1.19·10-s − 0.206·11-s + 0.962·12-s + 1.92·13-s − 0.617·14-s − 0.423·15-s + 0.110·16-s + 0.491·17-s + 0.544·18-s − 0.188·19-s − 1.22·20-s − 0.218·21-s − 0.337·22-s + 1.45·23-s + 0.628·24-s − 0.461·25-s + 3.13·26-s + 0.192·27-s − 0.629·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.821254623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.821254623\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 11 | \( 1 + 0.686T + 11T^{2} \) |
| 13 | \( 1 - 6.93T + 13T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 + 0.821T + 19T^{2} \) |
| 23 | \( 1 - 6.97T + 23T^{2} \) |
| 29 | \( 1 - 9.33T + 29T^{2} \) |
| 31 | \( 1 - 0.130T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 + 6.39T + 47T^{2} \) |
| 53 | \( 1 + 4.66T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 - 0.543T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 8.07T + 71T^{2} \) |
| 73 | \( 1 + 6.50T + 73T^{2} \) |
| 79 | \( 1 + 6.83T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 4.57T + 89T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−8.386114748670214064185959236898, −7.57313211376135394882431729444, −6.69467726652141770213167960637, −6.22162523779994200849644728297, −5.31507569880187844053873767440, −4.51405180141470691020267208198, −3.71728561637987906717790545240, −3.34539515864171246781175985118, −2.51474814832300977637175799430, −1.12775819090822462770118955410,
1.12775819090822462770118955410, 2.51474814832300977637175799430, 3.34539515864171246781175985118, 3.71728561637987906717790545240, 4.51405180141470691020267208198, 5.31507569880187844053873767440, 6.22162523779994200849644728297, 6.69467726652141770213167960637, 7.57313211376135394882431729444, 8.386114748670214064185959236898
