| L(s) = 1 | + 0.874·3-s + 2.82·7-s − 2.23·9-s + 0.763·11-s + 5.45·13-s + 7.40·17-s + 19-s + 2.47·21-s − 1.08·23-s − 4.57·27-s − 4.47·29-s + 4·31-s + 0.667·33-s + 2.62·37-s + 4.76·39-s − 6·41-s + 8.48·43-s + 8.48·47-s + 1.00·49-s + 6.47·51-s − 2.62·53-s + 0.874·57-s + 1.52·59-s − 11.7·61-s − 6.32·63-s + 11.1·67-s − 0.944·69-s + ⋯ |
| L(s) = 1 | + 0.504·3-s + 1.06·7-s − 0.745·9-s + 0.230·11-s + 1.51·13-s + 1.79·17-s + 0.229·19-s + 0.539·21-s − 0.225·23-s − 0.880·27-s − 0.830·29-s + 0.718·31-s + 0.116·33-s + 0.431·37-s + 0.762·39-s − 0.937·41-s + 1.29·43-s + 1.23·47-s + 0.142·49-s + 0.906·51-s − 0.360·53-s + 0.115·57-s + 0.198·59-s − 1.49·61-s − 0.796·63-s + 1.35·67-s − 0.113·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.254006852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.254006852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.874T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 - 7.40T + 17T^{2} \) |
| 23 | \( 1 + 1.08T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 2.62T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 5.24T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.953864138285862628615895768483, −7.49709343499266919492343572335, −6.36617761155203930113812792010, −5.71701990104330980455185118999, −5.22490293007781119226454854389, −4.13345410323861612041098135387, −3.54132336781312543575253479742, −2.77032592047010443397652452549, −1.70310051820595714157317184238, −0.956848719086101012015880938524,
0.956848719086101012015880938524, 1.70310051820595714157317184238, 2.77032592047010443397652452549, 3.54132336781312543575253479742, 4.13345410323861612041098135387, 5.22490293007781119226454854389, 5.71701990104330980455185118999, 6.36617761155203930113812792010, 7.49709343499266919492343572335, 7.953864138285862628615895768483
