| L(s) = 1 | + 2.58·2-s − 0.402·3-s + 4.69·4-s + 3.31·5-s − 1.04·6-s − 2.14·7-s + 6.97·8-s − 2.83·9-s + 8.57·10-s + 0.965·11-s − 1.89·12-s + 5.15·13-s − 5.54·14-s − 1.33·15-s + 8.66·16-s + 17-s − 7.34·18-s − 8.60·19-s + 15.5·20-s + 0.862·21-s + 2.49·22-s − 2.77·23-s − 2.80·24-s + 5.99·25-s + 13.3·26-s + 2.35·27-s − 10.0·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.232·3-s + 2.34·4-s + 1.48·5-s − 0.425·6-s − 0.809·7-s + 2.46·8-s − 0.945·9-s + 2.71·10-s + 0.291·11-s − 0.545·12-s + 1.42·13-s − 1.48·14-s − 0.344·15-s + 2.16·16-s + 0.242·17-s − 1.73·18-s − 1.97·19-s + 3.48·20-s + 0.188·21-s + 0.532·22-s − 0.578·23-s − 0.573·24-s + 1.19·25-s + 2.61·26-s + 0.452·27-s − 1.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.049326393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.049326393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 + 0.402T + 3T^{2} \) |
| 5 | \( 1 - 3.31T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 0.965T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 19 | \( 1 + 8.60T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 + 0.270T + 41T^{2} \) |
| 43 | \( 1 + 1.39T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 + 7.87T + 53T^{2} \) |
| 61 | \( 1 - 7.61T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 + 7.91T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 8.38T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−10.35940702308917917993097660749, −9.235221530848256878980837151513, −8.274603682934064711476552547946, −6.57911201901137988810038741800, −6.22950439563749668682480692020, −5.87459201981128627908633375881, −4.81531326962288841018944881975, −3.72045461273201547679165779949, −2.80564000592767869749211117019, −1.81484878643404107043076828721,
1.81484878643404107043076828721, 2.80564000592767869749211117019, 3.72045461273201547679165779949, 4.81531326962288841018944881975, 5.87459201981128627908633375881, 6.22950439563749668682480692020, 6.57911201901137988810038741800, 8.274603682934064711476552547946, 9.235221530848256878980837151513, 10.35940702308917917993097660749
