L(s) = 1 | − 2-s + 2.60·3-s + 4-s + 1.01·5-s − 2.60·6-s − 1.30·7-s − 8-s + 3.76·9-s − 1.01·10-s − 4.91·11-s + 2.60·12-s + 1.88·13-s + 1.30·14-s + 2.64·15-s + 16-s − 1.77·17-s − 3.76·18-s − 6.72·19-s + 1.01·20-s − 3.39·21-s + 4.91·22-s + 3.89·23-s − 2.60·24-s − 3.96·25-s − 1.88·26-s + 1.98·27-s − 1.30·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.50·3-s + 0.5·4-s + 0.454·5-s − 1.06·6-s − 0.493·7-s − 0.353·8-s + 1.25·9-s − 0.321·10-s − 1.48·11-s + 0.750·12-s + 0.522·13-s + 0.348·14-s + 0.682·15-s + 0.250·16-s − 0.430·17-s − 0.887·18-s − 1.54·19-s + 0.227·20-s − 0.740·21-s + 1.04·22-s + 0.812·23-s − 0.530·24-s − 0.793·25-s − 0.369·26-s + 0.382·27-s − 0.246·28-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 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 1.01T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 + 6.72T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 5.85T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + 1.16T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 - 5.74T + 61T^{2} \) |
| 67 | \( 1 + 4.91T + 67T^{2} \) |
| 71 | \( 1 + 2.27T + 71T^{2} \) |
| 73 | \( 1 - 4.06T + 73T^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 + 0.273T + 97T^{2} \) |
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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
−8.188076661975975415402840504622, −7.66432264489819433690707183116, −6.83147042943419810074204278992, −6.07272353488892275255586418001, −5.10263284107291702994283025595, −3.94436106517796089421325444123, −3.12687210657045987665392961153, −2.38590891131306228364900738243, −1.77189370799285264516078228087, 0,
1.77189370799285264516078228087, 2.38590891131306228364900738243, 3.12687210657045987665392961153, 3.94436106517796089421325444123, 5.10263284107291702994283025595, 6.07272353488892275255586418001, 6.83147042943419810074204278992, 7.66432264489819433690707183116, 8.188076661975975415402840504622