| L(s) = 1 | + 2-s + 3.26·3-s + 4-s + 0.0595·5-s + 3.26·6-s + 4.09·7-s + 8-s + 7.62·9-s + 0.0595·10-s − 4.44·11-s + 3.26·12-s + 0.622·13-s + 4.09·14-s + 0.194·15-s + 16-s − 3.36·17-s + 7.62·18-s − 2.76·19-s + 0.0595·20-s + 13.3·21-s − 4.44·22-s + 7.00·23-s + 3.26·24-s − 4.99·25-s + 0.622·26-s + 15.0·27-s + 4.09·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.88·3-s + 0.5·4-s + 0.0266·5-s + 1.33·6-s + 1.54·7-s + 0.353·8-s + 2.54·9-s + 0.0188·10-s − 1.34·11-s + 0.941·12-s + 0.172·13-s + 1.09·14-s + 0.0500·15-s + 0.250·16-s − 0.815·17-s + 1.79·18-s − 0.634·19-s + 0.0133·20-s + 2.91·21-s − 0.948·22-s + 1.46·23-s + 0.665·24-s − 0.999·25-s + 0.122·26-s + 2.90·27-s + 0.773·28-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
| \(L(1)\) |
\(\approx\) |
\(6.972343865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.972343865\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.26T + 3T^{2} \) |
| 5 | \( 1 - 0.0595T + 5T^{2} \) |
| 7 | \( 1 - 4.09T + 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 - 7.00T + 23T^{2} \) |
| 29 | \( 1 - 3.74T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 + 3.59T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 + 0.400T + 47T^{2} \) |
| 53 | \( 1 - 6.49T + 53T^{2} \) |
| 59 | \( 1 + 7.55T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 1.94T + 67T^{2} \) |
| 71 | \( 1 - 9.42T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 1.20T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 - 7.90T + 89T^{2} \) |
| 97 | \( 1 - 8.10T + 97T^{2} \) |
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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
−8.456801802503058742942489117477, −7.69294002034077911906218650910, −7.37740754497368911435214533819, −6.30290868807796754118628316017, −4.98511625129980967619962433853, −4.69376882533516994304655796419, −3.79486836962083498365381975356, −2.85889125959602345594200035082, −2.24493972205807543942703813381, −1.50643133859657322274361616831,
1.50643133859657322274361616831, 2.24493972205807543942703813381, 2.85889125959602345594200035082, 3.79486836962083498365381975356, 4.69376882533516994304655796419, 4.98511625129980967619962433853, 6.30290868807796754118628316017, 7.37740754497368911435214533819, 7.69294002034077911906218650910, 8.456801802503058742942489117477
