L(s) = 1 | − 3.38·3-s − 4.16·5-s + 7-s + 8.47·9-s − 11-s − 13-s + 14.1·15-s − 4.56·17-s − 0.578·19-s − 3.38·21-s − 6.33·23-s + 12.3·25-s − 18.5·27-s + 2.97·29-s + 5.14·31-s + 3.38·33-s − 4.16·35-s + 3.64·37-s + 3.38·39-s − 8.27·41-s − 10.1·43-s − 35.3·45-s − 3.73·47-s + 49-s + 15.4·51-s − 7.91·53-s + 4.16·55-s + ⋯ |
L(s) = 1 | − 1.95·3-s − 1.86·5-s + 0.377·7-s + 2.82·9-s − 0.301·11-s − 0.277·13-s + 3.64·15-s − 1.10·17-s − 0.132·19-s − 0.739·21-s − 1.32·23-s + 2.46·25-s − 3.57·27-s + 0.551·29-s + 0.924·31-s + 0.589·33-s − 0.703·35-s + 0.599·37-s + 0.542·39-s − 1.29·41-s − 1.55·43-s − 5.26·45-s − 0.545·47-s + 0.142·49-s + 2.16·51-s − 1.08·53-s + 0.561·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 3.38T + 3T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 + 0.578T + 19T^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 - 2.97T + 29T^{2} \) |
| 31 | \( 1 - 5.14T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 + 8.27T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 3.73T + 47T^{2} \) |
| 53 | \( 1 + 7.91T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 + 2.44T + 61T^{2} \) |
| 67 | \( 1 + 1.18T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 1.09T + 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.34979902046349258140028016234, −6.69877519881971587199602801162, −6.28028574504626011841372518774, −5.16649400424276325776230718502, −4.70803283823527116973990895040, −4.24386637902230813842579806039, −3.44794288695662002401175431837, −1.97372812087701460454186639777, −0.72222871217921031744753727822, 0,
0.72222871217921031744753727822, 1.97372812087701460454186639777, 3.44794288695662002401175431837, 4.24386637902230813842579806039, 4.70803283823527116973990895040, 5.16649400424276325776230718502, 6.28028574504626011841372518774, 6.69877519881971587199602801162, 7.34979902046349258140028016234