| L(s) = 1 | + 2.68·2-s + 4.33·3-s − 0.785·4-s − 2.08·5-s + 11.6·6-s − 24.9·7-s − 23.5·8-s − 8.23·9-s − 5.60·10-s + 3.82·11-s − 3.40·12-s + 17.6·13-s − 67.1·14-s − 9.03·15-s − 57.1·16-s − 22.1·18-s − 160.·19-s + 1.63·20-s − 108.·21-s + 10.2·22-s + 99.9·23-s − 102.·24-s − 120.·25-s + 47.4·26-s − 152.·27-s + 19.6·28-s + 200.·29-s + ⋯ |
| L(s) = 1 | + 0.949·2-s + 0.833·3-s − 0.0981·4-s − 0.186·5-s + 0.791·6-s − 1.34·7-s − 1.04·8-s − 0.305·9-s − 0.177·10-s + 0.104·11-s − 0.0818·12-s + 0.377·13-s − 1.28·14-s − 0.155·15-s − 0.892·16-s − 0.289·18-s − 1.94·19-s + 0.0183·20-s − 1.12·21-s + 0.0994·22-s + 0.906·23-s − 0.869·24-s − 0.965·25-s + 0.358·26-s − 1.08·27-s + 0.132·28-s + 1.28·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 2.68T + 8T^{2} \) |
| 3 | \( 1 - 4.33T + 27T^{2} \) |
| 5 | \( 1 + 2.08T + 125T^{2} \) |
| 7 | \( 1 + 24.9T + 343T^{2} \) |
| 11 | \( 1 - 3.82T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.6T + 2.19e3T^{2} \) |
| 19 | \( 1 + 160.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 76.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 468.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 96.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 707.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 304.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 470.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 717.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 367.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 903.T + 9.12e5T^{2} \) |
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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
−11.03573934610691369515757370714, −9.739996424191742620981601849510, −8.979793726548856708160399367995, −8.187413540566989237353926793991, −6.61547053761843112083050852265, −5.91453293801471401026200231247, −4.40000882258567329133347304482, −3.47836675348270245102783527031, −2.61661809863143040146612648359, 0,
2.61661809863143040146612648359, 3.47836675348270245102783527031, 4.40000882258567329133347304482, 5.91453293801471401026200231247, 6.61547053761843112083050852265, 8.187413540566989237353926793991, 8.979793726548856708160399367995, 9.739996424191742620981601849510, 11.03573934610691369515757370714
