L(s) = 1 | + 2-s − 0.724·3-s + 4-s − 3.73·5-s − 0.724·6-s − 3.63·7-s + 8-s − 2.47·9-s − 3.73·10-s + 5.50·11-s − 0.724·12-s + 4.62·13-s − 3.63·14-s + 2.70·15-s + 16-s + 4.53·17-s − 2.47·18-s − 6.26·19-s − 3.73·20-s + 2.63·21-s + 5.50·22-s + 4.08·23-s − 0.724·24-s + 8.95·25-s + 4.62·26-s + 3.96·27-s − 3.63·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.418·3-s + 0.5·4-s − 1.67·5-s − 0.295·6-s − 1.37·7-s + 0.353·8-s − 0.824·9-s − 1.18·10-s + 1.66·11-s − 0.209·12-s + 1.28·13-s − 0.971·14-s + 0.698·15-s + 0.250·16-s + 1.10·17-s − 0.583·18-s − 1.43·19-s − 0.835·20-s + 0.574·21-s + 1.17·22-s + 0.851·23-s − 0.147·24-s + 1.79·25-s + 0.906·26-s + 0.763·27-s − 0.686·28-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 | \( 1 + 0.724T + 3T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 - 5.50T + 11T^{2} \) |
| 13 | \( 1 - 4.62T + 13T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 19 | \( 1 + 6.26T + 19T^{2} \) |
| 23 | \( 1 - 4.08T + 23T^{2} \) |
| 29 | \( 1 - 9.49T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 1.21T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 + 0.133T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 9.32T + 67T^{2} \) |
| 71 | \( 1 + 3.23T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 9.34T + 83T^{2} \) |
| 89 | \( 1 - 9.11T + 89T^{2} \) |
| 97 | \( 1 + 1.38T + 97T^{2} \) |
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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
−8.105557430396076322287475641349, −7.05508413721670301032757868257, −6.41387678773837441342180859466, −6.15736802135835408931156466873, −4.95463998038702605636159789708, −4.05892441821637676298571224934, −3.45868094848625301451297759210, −3.11491697388243134239364328097, −1.23540827183363646882344111159, 0,
1.23540827183363646882344111159, 3.11491697388243134239364328097, 3.45868094848625301451297759210, 4.05892441821637676298571224934, 4.95463998038702605636159789708, 6.15736802135835408931156466873, 6.41387678773837441342180859466, 7.05508413721670301032757868257, 8.105557430396076322287475641349