L(s) = 1 | + 2-s − 2.69·3-s + 4-s + 2.56·5-s − 2.69·6-s − 4.70·7-s + 8-s + 4.28·9-s + 2.56·10-s − 3.35·11-s − 2.69·12-s − 1.77·13-s − 4.70·14-s − 6.91·15-s + 16-s + 7.86·17-s + 4.28·18-s + 2.18·19-s + 2.56·20-s + 12.6·21-s − 3.35·22-s + 1.70·23-s − 2.69·24-s + 1.56·25-s − 1.77·26-s − 3.48·27-s − 4.70·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.55·3-s + 0.5·4-s + 1.14·5-s − 1.10·6-s − 1.77·7-s + 0.353·8-s + 1.42·9-s + 0.810·10-s − 1.01·11-s − 0.779·12-s − 0.493·13-s − 1.25·14-s − 1.78·15-s + 0.250·16-s + 1.90·17-s + 1.01·18-s + 0.500·19-s + 0.572·20-s + 2.77·21-s − 0.716·22-s + 0.355·23-s − 0.551·24-s + 0.312·25-s − 0.348·26-s − 0.669·27-s − 0.888·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 + 1.77T + 13T^{2} \) |
| 17 | \( 1 - 7.86T + 17T^{2} \) |
| 19 | \( 1 - 2.18T + 19T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 - 0.397T + 31T^{2} \) |
| 37 | \( 1 - 0.252T + 37T^{2} \) |
| 41 | \( 1 + 6.51T + 41T^{2} \) |
| 43 | \( 1 - 9.23T + 43T^{2} \) |
| 47 | \( 1 + 0.749T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + 2.84T + 59T^{2} \) |
| 61 | \( 1 + 0.902T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 3.25T + 97T^{2} \) |
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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
−7.60833765218548653529072334047, −7.08319566422364502121551856094, −6.20739186747239175206100782319, −5.74253959680492938837338044244, −5.48130179308619944609286042164, −4.58023853255658606088243447389, −3.30856858799888617492304217351, −2.72271617681952282067758501198, −1.30572113957961668006216252844, 0,
1.30572113957961668006216252844, 2.72271617681952282067758501198, 3.30856858799888617492304217351, 4.58023853255658606088243447389, 5.48130179308619944609286042164, 5.74253959680492938837338044244, 6.20739186747239175206100782319, 7.08319566422364502121551856094, 7.60833765218548653529072334047