L(s) = 1 | − 2-s − 1.01·3-s + 4-s − 0.411·5-s + 1.01·6-s − 2.82·7-s − 8-s − 1.96·9-s + 0.411·10-s + 1.33·11-s − 1.01·12-s − 4.79·13-s + 2.82·14-s + 0.417·15-s + 16-s − 2.87·17-s + 1.96·18-s + 4.06·19-s − 0.411·20-s + 2.87·21-s − 1.33·22-s − 3.97·23-s + 1.01·24-s − 4.83·25-s + 4.79·26-s + 5.04·27-s − 2.82·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.586·3-s + 0.5·4-s − 0.183·5-s + 0.414·6-s − 1.06·7-s − 0.353·8-s − 0.656·9-s + 0.130·10-s + 0.401·11-s − 0.293·12-s − 1.33·13-s + 0.756·14-s + 0.107·15-s + 0.250·16-s − 0.698·17-s + 0.463·18-s + 0.932·19-s − 0.0919·20-s + 0.627·21-s − 0.283·22-s − 0.828·23-s + 0.207·24-s − 0.966·25-s + 0.940·26-s + 0.971·27-s − 0.534·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2267851516\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2267851516\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 + 1.01T + 3T^{2} \) |
| 5 | \( 1 + 0.411T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 1.33T + 11T^{2} \) |
| 13 | \( 1 + 4.79T + 13T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 + 0.335T + 29T^{2} \) |
| 31 | \( 1 - 0.459T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 0.433T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 3.83T + 53T^{2} \) |
| 59 | \( 1 + 9.99T + 59T^{2} \) |
| 61 | \( 1 + 1.34T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 0.0392T + 71T^{2} \) |
| 73 | \( 1 - 5.14T + 73T^{2} \) |
| 79 | \( 1 - 6.86T + 79T^{2} \) |
| 83 | \( 1 + 5.20T + 83T^{2} \) |
| 89 | \( 1 + 6.43T + 89T^{2} \) |
| 97 | \( 1 - 2.23T + 97T^{2} \) |
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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
−8.448440264952547638624244961981, −7.74803584270069498290162499319, −6.87078726569953199679447724235, −6.45432145876094012398766869012, −5.60358003016343736777323738048, −4.86081521277676482840749094738, −3.65875409040447037095354479459, −2.89754017500172830526982814286, −1.87196887447329131903425421670, −0.29408151696601778671310418384,
0.29408151696601778671310418384, 1.87196887447329131903425421670, 2.89754017500172830526982814286, 3.65875409040447037095354479459, 4.86081521277676482840749094738, 5.60358003016343736777323738048, 6.45432145876094012398766869012, 6.87078726569953199679447724235, 7.74803584270069498290162499319, 8.448440264952547638624244961981