| L(s) = 1 | + 2.90·3-s + 3.85·5-s + 7-s + 5.44·9-s − 1.30·11-s − 1.59·13-s + 11.2·15-s + 1.50·17-s + 2.06·19-s + 2.90·21-s + 4.77·23-s + 9.88·25-s + 7.10·27-s − 5.79·29-s + 4.10·31-s − 3.77·33-s + 3.85·35-s − 2.34·37-s − 4.62·39-s + 7.45·41-s − 8.04·43-s + 21.0·45-s − 3.59·47-s + 49-s + 4.38·51-s + 0.955·53-s − 5.01·55-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 1.72·5-s + 0.377·7-s + 1.81·9-s − 0.392·11-s − 0.441·13-s + 2.89·15-s + 0.365·17-s + 0.474·19-s + 0.634·21-s + 0.994·23-s + 1.97·25-s + 1.36·27-s − 1.07·29-s + 0.738·31-s − 0.657·33-s + 0.652·35-s − 0.384·37-s − 0.741·39-s + 1.16·41-s − 1.22·43-s + 3.13·45-s − 0.524·47-s + 0.142·49-s + 0.614·51-s + 0.131·53-s − 0.676·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.983471723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.983471723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 13 | \( 1 + 1.59T + 13T^{2} \) |
| 17 | \( 1 - 1.50T + 17T^{2} \) |
| 19 | \( 1 - 2.06T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 + 5.79T + 29T^{2} \) |
| 31 | \( 1 - 4.10T + 31T^{2} \) |
| 37 | \( 1 + 2.34T + 37T^{2} \) |
| 41 | \( 1 - 7.45T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 - 0.955T + 53T^{2} \) |
| 59 | \( 1 - 1.60T + 59T^{2} \) |
| 61 | \( 1 - 4.55T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 1.77T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 8.45T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.998217275027597860955808814730, −7.34784583861284526021995738277, −6.71183581632492753338709751955, −5.70531362756731342155448850527, −5.15671585143453666670810159536, −4.30044575263141734472537471420, −3.17238575996410312980923416326, −2.70998191455122757049127614536, −1.94772220840727728796996861648, −1.29863647657838211136972972191,
1.29863647657838211136972972191, 1.94772220840727728796996861648, 2.70998191455122757049127614536, 3.17238575996410312980923416326, 4.30044575263141734472537471420, 5.15671585143453666670810159536, 5.70531362756731342155448850527, 6.71183581632492753338709751955, 7.34784583861284526021995738277, 7.998217275027597860955808814730
