L(s) = 1 | − 2-s − 0.779·3-s + 4-s − 0.867·5-s + 0.779·6-s − 3.45·7-s − 8-s − 2.39·9-s + 0.867·10-s + 3.23·11-s − 0.779·12-s + 0.0959·13-s + 3.45·14-s + 0.675·15-s + 16-s + 5.34·17-s + 2.39·18-s + 1.96·19-s − 0.867·20-s + 2.68·21-s − 3.23·22-s − 4.56·23-s + 0.779·24-s − 4.24·25-s − 0.0959·26-s + 4.20·27-s − 3.45·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.449·3-s + 0.5·4-s − 0.387·5-s + 0.318·6-s − 1.30·7-s − 0.353·8-s − 0.797·9-s + 0.274·10-s + 0.975·11-s − 0.224·12-s + 0.0266·13-s + 0.922·14-s + 0.174·15-s + 0.250·16-s + 1.29·17-s + 0.563·18-s + 0.451·19-s − 0.193·20-s + 0.586·21-s − 0.689·22-s − 0.952·23-s + 0.159·24-s − 0.849·25-s − 0.0188·26-s + 0.808·27-s − 0.652·28-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 | \( 1 + 0.779T + 3T^{2} \) |
| 5 | \( 1 + 0.867T + 5T^{2} \) |
| 7 | \( 1 + 3.45T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 0.0959T + 13T^{2} \) |
| 17 | \( 1 - 5.34T + 17T^{2} \) |
| 19 | \( 1 - 1.96T + 19T^{2} \) |
| 23 | \( 1 + 4.56T + 23T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 - 0.192T + 31T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 - 6.89T + 41T^{2} \) |
| 43 | \( 1 + 3.27T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 1.21T + 53T^{2} \) |
| 59 | \( 1 - 0.581T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 - 9.49T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 2.62T + 79T^{2} \) |
| 83 | \( 1 - 7.45T + 83T^{2} \) |
| 89 | \( 1 - 7.41T + 89T^{2} \) |
| 97 | \( 1 + 2.66T + 97T^{2} \) |
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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
−7.991381328541326233605776879834, −7.48796945274159000627019642962, −6.53025953473652812097039004883, −6.02608765862588988407568396142, −5.42964797675787976340856748990, −3.97339230492411582287335797951, −3.43272333389163458106486010647, −2.46696880481907368147987917389, −1.05458383713019572967594785420, 0,
1.05458383713019572967594785420, 2.46696880481907368147987917389, 3.43272333389163458106486010647, 3.97339230492411582287335797951, 5.42964797675787976340856748990, 6.02608765862588988407568396142, 6.53025953473652812097039004883, 7.48796945274159000627019642962, 7.991381328541326233605776879834