| L(s) = 1 | + 2.16·3-s + 4.16·5-s + 0.683·7-s + 1.68·9-s + 11-s + 2.68·13-s + 9.01·15-s + 3.36·17-s + 19-s + 1.48·21-s + 1.36·23-s + 12.3·25-s − 2.84·27-s + 2.79·29-s − 5.01·31-s + 2.16·33-s + 2.84·35-s − 11.6·37-s + 5.80·39-s − 7.01·41-s − 7.86·43-s + 7.01·45-s − 2.96·47-s − 6.53·49-s + 7.28·51-s − 10.3·53-s + 4.16·55-s + ⋯ |
| L(s) = 1 | + 1.24·3-s + 1.86·5-s + 0.258·7-s + 0.561·9-s + 0.301·11-s + 0.744·13-s + 2.32·15-s + 0.816·17-s + 0.229·19-s + 0.323·21-s + 0.285·23-s + 2.46·25-s − 0.548·27-s + 0.519·29-s − 0.900·31-s + 0.376·33-s + 0.481·35-s − 1.92·37-s + 0.930·39-s − 1.09·41-s − 1.19·43-s + 1.04·45-s − 0.431·47-s − 0.933·49-s + 1.02·51-s − 1.41·53-s + 0.561·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.572184190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.572184190\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 - 0.683T + 7T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 + 5.01T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 7.01T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 + 2.96T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 9.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 0.683T + 67T^{2} \) |
| 71 | \( 1 + 3.53T + 71T^{2} \) |
| 73 | \( 1 - 0.407T + 73T^{2} \) |
| 79 | \( 1 - 7.28T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.614413052012252501474974336119, −8.226413851520451301351464101999, −7.06816231679602697992828203407, −6.43647303648358285892791627248, −5.53691470890488998311725912715, −4.98641330874730776159381176203, −3.56019209935327626635717376663, −3.03720168481831504883657897928, −1.92030383960385532509177307004, −1.47323712312371050661870406750,
1.47323712312371050661870406750, 1.92030383960385532509177307004, 3.03720168481831504883657897928, 3.56019209935327626635717376663, 4.98641330874730776159381176203, 5.53691470890488998311725912715, 6.43647303648358285892791627248, 7.06816231679602697992828203407, 8.226413851520451301351464101999, 8.614413052012252501474974336119
