| L(s) = 1 | − 2.41·2-s − 2.40·3-s + 3.82·4-s + 5-s + 5.79·6-s + 1.40·7-s − 4.40·8-s + 2.76·9-s − 2.41·10-s + 1.34·11-s − 9.18·12-s + 2.31·13-s − 3.38·14-s − 2.40·15-s + 2.98·16-s + 7.01·17-s − 6.68·18-s − 3.78·19-s + 3.82·20-s − 3.37·21-s − 3.25·22-s − 4.96·23-s + 10.5·24-s + 25-s − 5.59·26-s + 0.554·27-s + 5.37·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 1.38·3-s + 1.91·4-s + 0.447·5-s + 2.36·6-s + 0.530·7-s − 1.55·8-s + 0.923·9-s − 0.763·10-s + 0.406·11-s − 2.65·12-s + 0.642·13-s − 0.905·14-s − 0.620·15-s + 0.746·16-s + 1.70·17-s − 1.57·18-s − 0.869·19-s + 0.855·20-s − 0.735·21-s − 0.693·22-s − 1.03·23-s + 2.16·24-s + 0.200·25-s − 1.09·26-s + 0.106·27-s + 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5961876258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5961876258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1601 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2.40T + 3T^{2} \) |
| 7 | \( 1 - 1.40T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 - 2.31T + 13T^{2} \) |
| 17 | \( 1 - 7.01T + 17T^{2} \) |
| 19 | \( 1 + 3.78T + 19T^{2} \) |
| 23 | \( 1 + 4.96T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 - 4.17T + 37T^{2} \) |
| 41 | \( 1 + 2.56T + 41T^{2} \) |
| 43 | \( 1 + 2.91T + 43T^{2} \) |
| 47 | \( 1 + 0.0759T + 47T^{2} \) |
| 53 | \( 1 + 8.51T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 + 0.548T + 97T^{2} \) |
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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
−8.058755771787132090461976939411, −7.21972289179864218385130225513, −6.38810779426552622504560968611, −6.10401494223574701482133815913, −5.33469636686768065818333888375, −4.47987858685045107556279170688, −3.35586135329553801258925593805, −2.05560712228287595514986799228, −1.36791815701121466224448582544, −0.59428022695254931709777825436,
0.59428022695254931709777825436, 1.36791815701121466224448582544, 2.05560712228287595514986799228, 3.35586135329553801258925593805, 4.47987858685045107556279170688, 5.33469636686768065818333888375, 6.10401494223574701482133815913, 6.38810779426552622504560968611, 7.21972289179864218385130225513, 8.058755771787132090461976939411
