| L(s) = 1 | − 0.171i·5-s + 5.48·7-s + 3.34i·11-s + 14.9·13-s − 17.3i·17-s + 24.9·19-s + 40.2i·23-s + 24.9·25-s − 42i·29-s + 7.48·31-s − 0.941i·35-s − 19.0·37-s + 38.9i·41-s − 30.9·43-s + 8.05i·47-s + ⋯ |
| L(s) = 1 | − 0.0343i·5-s + 0.783·7-s + 0.303i·11-s + 1.15·13-s − 1.01i·17-s + 1.31·19-s + 1.75i·23-s + 0.998·25-s − 1.44i·29-s + 0.241·31-s − 0.0268i·35-s − 0.514·37-s + 0.949i·41-s − 0.720·43-s + 0.171i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.74681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74681\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.171iT - 25T^{2} \) |
| 7 | \( 1 - 5.48T + 49T^{2} \) |
| 11 | \( 1 - 3.34iT - 121T^{2} \) |
| 13 | \( 1 - 14.9T + 169T^{2} \) |
| 17 | \( 1 + 17.3iT - 289T^{2} \) |
| 19 | \( 1 - 24.9T + 361T^{2} \) |
| 23 | \( 1 - 40.2iT - 529T^{2} \) |
| 29 | \( 1 + 42iT - 841T^{2} \) |
| 31 | \( 1 - 7.48T + 961T^{2} \) |
| 37 | \( 1 + 19.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 30.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 8.05iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 99.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 97.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 74.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 46.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 121.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 107.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 29.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 78.8T + 9.40e3T^{2} \) |
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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
−11.68465255994789095303356040069, −11.47224770490993414466977834993, −10.10530966358190000591169800659, −9.157211068602344675951273232872, −8.051348791001884862673035726873, −7.14755742874091157850481962171, −5.74070309820807726741042242165, −4.69668050757654793678750660705, −3.22031679948691767909826366995, −1.36857783633488885672175505706,
1.36499465014207392076189423723, 3.23978817656453217055289039956, 4.63827959934901184803169099348, 5.83879449562325201496636663283, 7.00009272728732160845430687009, 8.304520835761411012804160708155, 8.869295831858358771848985450471, 10.43778055343281617035005164492, 10.98082740228126007456983174952, 12.07848668493423303697317165245
