L(s) = 1 | − 1.05·2-s − 3-s − 0.883·4-s + 1.53·5-s + 1.05·6-s − 3.09·7-s + 3.04·8-s + 9-s − 1.61·10-s − 11-s + 0.883·12-s + 3.38·13-s + 3.26·14-s − 1.53·15-s − 1.45·16-s − 3.57·17-s − 1.05·18-s + 1.31·19-s − 1.35·20-s + 3.09·21-s + 1.05·22-s − 5.12·23-s − 3.04·24-s − 2.65·25-s − 3.57·26-s − 27-s + 2.73·28-s + ⋯ |
L(s) = 1 | − 0.747·2-s − 0.577·3-s − 0.441·4-s + 0.684·5-s + 0.431·6-s − 1.16·7-s + 1.07·8-s + 0.333·9-s − 0.511·10-s − 0.301·11-s + 0.254·12-s + 0.938·13-s + 0.873·14-s − 0.395·15-s − 0.363·16-s − 0.867·17-s − 0.249·18-s + 0.301·19-s − 0.302·20-s + 0.674·21-s + 0.225·22-s − 1.06·23-s − 0.621·24-s − 0.531·25-s − 0.701·26-s − 0.192·27-s + 0.516·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 1.05T + 2T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 + 3.09T + 7T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 3.57T + 17T^{2} \) |
| 19 | \( 1 - 1.31T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 - 6.01T + 31T^{2} \) |
| 37 | \( 1 - 5.49T + 37T^{2} \) |
| 41 | \( 1 - 8.30T + 41T^{2} \) |
| 43 | \( 1 + 6.40T + 43T^{2} \) |
| 47 | \( 1 - 4.95T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 4.40T + 59T^{2} \) |
| 67 | \( 1 - 3.91T + 67T^{2} \) |
| 71 | \( 1 + 6.45T + 71T^{2} \) |
| 73 | \( 1 + 9.63T + 73T^{2} \) |
| 79 | \( 1 + 2.79T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.869939492482547282690486857261, −8.171124547088595776216638568880, −7.20587331172839362782326899784, −6.22715933256345311458082516261, −5.89437431419828095568862102184, −4.67774699315458175970810449131, −3.89462934865076389730004165026, −2.58734668803053703886646186341, −1.25231657983292375202851797777, 0,
1.25231657983292375202851797777, 2.58734668803053703886646186341, 3.89462934865076389730004165026, 4.67774699315458175970810449131, 5.89437431419828095568862102184, 6.22715933256345311458082516261, 7.20587331172839362782326899784, 8.171124547088595776216638568880, 8.869939492482547282690486857261