L(s) = 1 | + 1.88·2-s + 3-s + 1.56·4-s + 1.88·6-s − 1.44·7-s − 0.827·8-s + 9-s + 0.859·11-s + 1.56·12-s − 2.84·13-s − 2.73·14-s − 4.68·16-s − 5.63·17-s + 1.88·18-s − 6.59·19-s − 1.44·21-s + 1.62·22-s + 0.190·23-s − 0.827·24-s − 5.36·26-s + 27-s − 2.26·28-s − 10.3·29-s + 7.02·31-s − 7.18·32-s + 0.859·33-s − 10.6·34-s + ⋯ |
L(s) = 1 | + 1.33·2-s + 0.577·3-s + 0.780·4-s + 0.770·6-s − 0.547·7-s − 0.292·8-s + 0.333·9-s + 0.259·11-s + 0.450·12-s − 0.788·13-s − 0.730·14-s − 1.17·16-s − 1.36·17-s + 0.444·18-s − 1.51·19-s − 0.316·21-s + 0.345·22-s + 0.0396·23-s − 0.168·24-s − 1.05·26-s + 0.192·27-s − 0.427·28-s − 1.93·29-s + 1.26·31-s − 1.27·32-s + 0.149·33-s − 1.82·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 - 0.859T + 11T^{2} \) |
| 13 | \( 1 + 2.84T + 13T^{2} \) |
| 17 | \( 1 + 5.63T + 17T^{2} \) |
| 19 | \( 1 + 6.59T + 19T^{2} \) |
| 23 | \( 1 - 0.190T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 - 5.67T + 37T^{2} \) |
| 41 | \( 1 + 8.12T + 41T^{2} \) |
| 43 | \( 1 - 4.91T + 43T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 0.840T + 71T^{2} \) |
| 73 | \( 1 + 4.72T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 0.110T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 1.88T + 97T^{2} \) |
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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
−8.240846077172338336073516967627, −7.15393264866188901222768555951, −6.57744315438666070532780284969, −5.94537208745276440307276201061, −4.91788960479914538919372682268, −4.27379945277529862627027303781, −3.67955892585155909579817648475, −2.66090183900022310044033050815, −2.08205227200765841338269729819, 0,
2.08205227200765841338269729819, 2.66090183900022310044033050815, 3.67955892585155909579817648475, 4.27379945277529862627027303781, 4.91788960479914538919372682268, 5.94537208745276440307276201061, 6.57744315438666070532780284969, 7.15393264866188901222768555951, 8.240846077172338336073516967627