| L(s) = 1 | + 2-s + 1.01·3-s + 4-s − 2.75·5-s + 1.01·6-s − 1.44·7-s + 8-s − 1.96·9-s − 2.75·10-s + 0.884·11-s + 1.01·12-s + 2.70·13-s − 1.44·14-s − 2.80·15-s + 16-s + 3.48·17-s − 1.96·18-s + 3.44·19-s − 2.75·20-s − 1.46·21-s + 0.884·22-s − 7.98·23-s + 1.01·24-s + 2.59·25-s + 2.70·26-s − 5.05·27-s − 1.44·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.588·3-s + 0.5·4-s − 1.23·5-s + 0.415·6-s − 0.545·7-s + 0.353·8-s − 0.654·9-s − 0.871·10-s + 0.266·11-s + 0.294·12-s + 0.750·13-s − 0.385·14-s − 0.724·15-s + 0.250·16-s + 0.844·17-s − 0.462·18-s + 0.789·19-s − 0.616·20-s − 0.320·21-s + 0.188·22-s − 1.66·23-s + 0.207·24-s + 0.518·25-s + 0.530·26-s − 0.972·27-s − 0.272·28-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 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 + 2.75T + 5T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 - 0.884T + 11T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 19 | \( 1 - 3.44T + 19T^{2} \) |
| 23 | \( 1 + 7.98T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 9.84T + 31T^{2} \) |
| 37 | \( 1 - 8.68T + 37T^{2} \) |
| 41 | \( 1 + 4.84T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 + 0.952T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 0.0386T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 + 5.18T + 79T^{2} \) |
| 83 | \( 1 - 1.54T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−7.973024156836622813757755550328, −7.58261029893927870843557829791, −6.43808881599502305266300320274, −5.95735062169904881258922891955, −4.96558172988569612690911761050, −3.95853527834919859660697462699, −3.48133200030881708561256462310, −2.93521800120406100972029960047, −1.60212136928246107297496575197, 0,
1.60212136928246107297496575197, 2.93521800120406100972029960047, 3.48133200030881708561256462310, 3.95853527834919859660697462699, 4.96558172988569612690911761050, 5.95735062169904881258922891955, 6.43808881599502305266300320274, 7.58261029893927870843557829791, 7.973024156836622813757755550328
