| L(s) = 1 | − 2.62·2-s − 2.12·3-s + 4.87·4-s + 2.57·5-s + 5.56·6-s − 3.69·7-s − 7.54·8-s + 1.50·9-s − 6.75·10-s − 0.495·11-s − 10.3·12-s + 5.97·13-s + 9.68·14-s − 5.46·15-s + 10.0·16-s − 3.51·17-s − 3.95·18-s − 8.34·19-s + 12.5·20-s + 7.84·21-s + 1.29·22-s − 5.47·23-s + 16.0·24-s + 1.62·25-s − 15.6·26-s + 3.16·27-s − 18.0·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.22·3-s + 2.43·4-s + 1.15·5-s + 2.27·6-s − 1.39·7-s − 2.66·8-s + 0.502·9-s − 2.13·10-s − 0.149·11-s − 2.98·12-s + 1.65·13-s + 2.58·14-s − 1.41·15-s + 2.50·16-s − 0.853·17-s − 0.932·18-s − 1.91·19-s + 2.80·20-s + 1.71·21-s + 0.276·22-s − 1.14·23-s + 3.27·24-s + 0.325·25-s − 3.07·26-s + 0.609·27-s − 3.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2178531023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2178531023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 2.12T + 3T^{2} \) |
| 5 | \( 1 - 2.57T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 0.495T + 11T^{2} \) |
| 13 | \( 1 - 5.97T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 + 8.34T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 - 6.95T + 29T^{2} \) |
| 31 | \( 1 - 1.64T + 31T^{2} \) |
| 37 | \( 1 - 0.114T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 6.50T + 59T^{2} \) |
| 61 | \( 1 + 1.38T + 61T^{2} \) |
| 67 | \( 1 + 5.99T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 0.963T + 79T^{2} \) |
| 83 | \( 1 + 8.20T + 83T^{2} \) |
| 89 | \( 1 + 1.76T + 89T^{2} \) |
| 97 | \( 1 + 1.36T + 97T^{2} \) |
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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
−8.158402443477887057549556199123, −6.70472960269811994597265144903, −6.55347179456621067803139007739, −6.20222187202562953308260057757, −5.63703788526210319603799268531, −4.31825500790604524955532702737, −3.13021608606853481026098982016, −2.22237063370530443641690543168, −1.45303869705565359071172798262, −0.33195653020719694984290311336,
0.33195653020719694984290311336, 1.45303869705565359071172798262, 2.22237063370530443641690543168, 3.13021608606853481026098982016, 4.31825500790604524955532702737, 5.63703788526210319603799268531, 6.20222187202562953308260057757, 6.55347179456621067803139007739, 6.70472960269811994597265144903, 8.158402443477887057549556199123
