| L(s) = 1 | − 2.25·2-s − 1.69·3-s + 3.07·4-s − 1.12·5-s + 3.82·6-s − 4.02·7-s − 2.41·8-s − 0.122·9-s + 2.52·10-s − 2.48·11-s − 5.20·12-s + 4.38·13-s + 9.06·14-s + 1.90·15-s − 0.711·16-s − 3.94·17-s + 0.275·18-s + 19-s − 3.43·20-s + 6.83·21-s + 5.59·22-s − 3.68·23-s + 4.09·24-s − 3.74·25-s − 9.86·26-s + 5.29·27-s − 12.3·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.979·3-s + 1.53·4-s − 0.500·5-s + 1.55·6-s − 1.52·7-s − 0.852·8-s − 0.0407·9-s + 0.797·10-s − 0.749·11-s − 1.50·12-s + 1.21·13-s + 2.42·14-s + 0.490·15-s − 0.177·16-s − 0.957·17-s + 0.0648·18-s + 0.229·19-s − 0.769·20-s + 1.49·21-s + 1.19·22-s − 0.768·23-s + 0.834·24-s − 0.749·25-s − 1.93·26-s + 1.01·27-s − 2.33·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 3 | \( 1 + 1.69T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 4.02T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 17 | \( 1 + 3.94T + 17T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 - 3.83T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 - 6.52T + 37T^{2} \) |
| 41 | \( 1 + 4.13T + 41T^{2} \) |
| 43 | \( 1 + 4.23T + 43T^{2} \) |
| 47 | \( 1 - 7.04T + 47T^{2} \) |
| 53 | \( 1 - 2.97T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 8.71T + 67T^{2} \) |
| 71 | \( 1 - 9.49T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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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
−8.195635037548835952042707198486, −7.48408006585941790721959258604, −6.48357383714993008864057950934, −6.35552746023865000924696430817, −5.39624802233050183710389751006, −4.18405607594017609131775637618, −3.23056141551002201434844749461, −2.19392036308005870328828597193, −0.73821434393454053724814499489, 0,
0.73821434393454053724814499489, 2.19392036308005870328828597193, 3.23056141551002201434844749461, 4.18405607594017609131775637618, 5.39624802233050183710389751006, 6.35552746023865000924696430817, 6.48357383714993008864057950934, 7.48408006585941790721959258604, 8.195635037548835952042707198486
