| L(s) = 1 | + 2-s + 4-s + 5-s − 5.07·7-s + 8-s + 10-s − 1.99·11-s − 5.82·13-s − 5.07·14-s + 16-s + 4.54·17-s − 7.38·19-s + 20-s − 1.99·22-s + 1.23·23-s + 25-s − 5.82·26-s − 5.07·28-s − 2.25·29-s − 2.32·31-s + 32-s + 4.54·34-s − 5.07·35-s + 8.84·37-s − 7.38·38-s + 40-s + 10.3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.91·7-s + 0.353·8-s + 0.316·10-s − 0.601·11-s − 1.61·13-s − 1.35·14-s + 0.250·16-s + 1.10·17-s − 1.69·19-s + 0.223·20-s − 0.425·22-s + 0.258·23-s + 0.200·25-s − 1.14·26-s − 0.959·28-s − 0.419·29-s − 0.417·31-s + 0.176·32-s + 0.778·34-s − 0.857·35-s + 1.45·37-s − 1.19·38-s + 0.158·40-s + 1.62·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.864126147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864126147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 7 | \( 1 + 5.07T + 7T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 + 5.82T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 + 7.38T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 - 8.84T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 3.00T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 8.65T + 71T^{2} \) |
| 73 | \( 1 - 1.77T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 9.44T + 83T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.52439065428947524942403852988, −7.03719493660999430047250653331, −6.34430129371547822600403255760, −5.77130852324223863558016506137, −5.15428376817581919935135148681, −4.22163053602871753268036585255, −3.50080454394411223382386086126, −2.61206074849167186731584996106, −2.29727077567748943852450214580, −0.56216829394299677415453076831,
0.56216829394299677415453076831, 2.29727077567748943852450214580, 2.61206074849167186731584996106, 3.50080454394411223382386086126, 4.22163053602871753268036585255, 5.15428376817581919935135148681, 5.77130852324223863558016506137, 6.34430129371547822600403255760, 7.03719493660999430047250653331, 7.52439065428947524942403852988
