| L(s) = 1 | − 11.5·3-s + 27.0·7-s − 110.·9-s − 226.·11-s − 511.·13-s − 387.·17-s + 1.33e3·19-s − 311.·21-s + 545.·23-s + 4.07e3·27-s − 4.63e3·29-s − 2.99e3·31-s + 2.60e3·33-s + 1.26e3·37-s + 5.89e3·39-s − 1.71e4·41-s − 1.65e4·43-s + 1.30e4·47-s − 1.60e4·49-s + 4.46e3·51-s + 2.89e4·53-s − 1.53e4·57-s + 3.44e4·59-s − 2.41e4·61-s − 2.98e3·63-s + 2.93e4·67-s − 6.28e3·69-s + ⋯ |
| L(s) = 1 | − 0.739·3-s + 0.208·7-s − 0.453·9-s − 0.563·11-s − 0.839·13-s − 0.325·17-s + 0.848·19-s − 0.154·21-s + 0.214·23-s + 1.07·27-s − 1.02·29-s − 0.559·31-s + 0.416·33-s + 0.151·37-s + 0.620·39-s − 1.59·41-s − 1.36·43-s + 0.860·47-s − 0.956·49-s + 0.240·51-s + 1.41·53-s − 0.627·57-s + 1.28·59-s − 0.830·61-s − 0.0946·63-s + 0.799·67-s − 0.158·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9492186789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9492186789\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 11.5T + 243T^{2} \) |
| 7 | \( 1 - 27.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 226.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 511.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 387.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 545.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.89e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.06e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.01e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.05e4T + 8.58e9T^{2} \) |
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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
−10.56756038960455007365553590237, −9.674978240046098165215528507630, −8.610956995624248792019258833458, −7.58912611084731348858165391768, −6.65945158157963821284041533125, −5.45226657850375047959547540479, −4.95556458544384575079629690862, −3.40816096822972329998766620377, −2.11659886807368425994540263481, −0.51508641560616226091748210157,
0.51508641560616226091748210157, 2.11659886807368425994540263481, 3.40816096822972329998766620377, 4.95556458544384575079629690862, 5.45226657850375047959547540479, 6.65945158157963821284041533125, 7.58912611084731348858165391768, 8.610956995624248792019258833458, 9.674978240046098165215528507630, 10.56756038960455007365553590237
