| L(s) = 1 | − 1.58·2-s − 3·3-s − 5.49·4-s − 18.8·5-s + 4.74·6-s + 29.4·7-s + 21.3·8-s + 9·9-s + 29.9·10-s + 25.6·11-s + 16.4·12-s − 8.38·13-s − 46.6·14-s + 56.6·15-s + 10.1·16-s − 70.0·17-s − 14.2·18-s + 48.9·19-s + 103.·20-s − 88.3·21-s − 40.6·22-s + 107.·23-s − 64.0·24-s + 231.·25-s + 13.2·26-s − 27·27-s − 161.·28-s + ⋯ |
| L(s) = 1 | − 0.559·2-s − 0.577·3-s − 0.686·4-s − 1.68·5-s + 0.323·6-s + 1.59·7-s + 0.944·8-s + 0.333·9-s + 0.945·10-s + 0.703·11-s + 0.396·12-s − 0.178·13-s − 0.890·14-s + 0.975·15-s + 0.158·16-s − 0.999·17-s − 0.186·18-s + 0.590·19-s + 1.16·20-s − 0.918·21-s − 0.393·22-s + 0.971·23-s − 0.545·24-s + 1.85·25-s + 0.100·26-s − 0.192·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6738582040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6738582040\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 1.58T + 8T^{2} \) |
| 5 | \( 1 + 18.8T + 125T^{2} \) |
| 7 | \( 1 - 29.4T + 343T^{2} \) |
| 11 | \( 1 - 25.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.38T + 2.19e3T^{2} \) |
| 17 | \( 1 + 70.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 107.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 330.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.51T + 6.89e4T^{2} \) |
| 43 | \( 1 - 424.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 77.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 584.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 427.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 248.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 925.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 738.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 391.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.40e3T + 9.12e5T^{2} \) |
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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
−13.79183651860018705752976297412, −12.23173670794993757786980184606, −11.45198564942509391371494944382, −10.70587469119614915637106968224, −8.974385377563786926406719682192, −8.096423537590551831984562368685, −7.19208186387962333199664067448, −4.89844256104097052945327673825, −4.17246947013417899360841638441, −0.897521062195434883324017233483,
0.897521062195434883324017233483, 4.17246947013417899360841638441, 4.89844256104097052945327673825, 7.19208186387962333199664067448, 8.096423537590551831984562368685, 8.974385377563786926406719682192, 10.70587469119614915637106968224, 11.45198564942509391371494944382, 12.23173670794993757786980184606, 13.79183651860018705752976297412
