| L(s) = 1 | + 1.26·2-s + 6.09·3-s − 6.40·4-s − 13.2·5-s + 7.69·6-s + 27.2·7-s − 18.1·8-s + 10.1·9-s − 16.7·10-s − 45.5·11-s − 39.0·12-s − 52.5·13-s + 34.3·14-s − 80.8·15-s + 28.3·16-s + 12.8·18-s + 3.08·19-s + 84.9·20-s + 166.·21-s − 57.4·22-s − 112.·23-s − 110.·24-s + 50.8·25-s − 66.2·26-s − 102.·27-s − 174.·28-s + 18.6·29-s + ⋯ |
| L(s) = 1 | + 0.446·2-s + 1.17·3-s − 0.801·4-s − 1.18·5-s + 0.523·6-s + 1.47·7-s − 0.803·8-s + 0.376·9-s − 0.529·10-s − 1.24·11-s − 0.939·12-s − 1.12·13-s + 0.656·14-s − 1.39·15-s + 0.442·16-s + 0.167·18-s + 0.0372·19-s + 0.950·20-s + 1.72·21-s − 0.556·22-s − 1.01·23-s − 0.942·24-s + 0.407·25-s − 0.499·26-s − 0.731·27-s − 1.17·28-s + 0.119·29-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 | \( 1 - 1.26T + 8T^{2} \) |
| 3 | \( 1 - 6.09T + 27T^{2} \) |
| 5 | \( 1 + 13.2T + 125T^{2} \) |
| 7 | \( 1 - 27.2T + 343T^{2} \) |
| 11 | \( 1 + 45.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 3.08T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 238.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 383.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 468.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 105.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 586.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 401.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 481.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 725.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 382.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 182.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 623.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 369.T + 9.12e5T^{2} \) |
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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
−11.03981422853347132469563580569, −9.783407111288052623996500571699, −8.698370701982385061548709054403, −7.87598117685228312216412779573, −7.68868932990873702715231737160, −5.37529566414826305905694816024, −4.55273338989641640076751039960, −3.58511338042728606061281075872, −2.30686478130574781048998629109, 0,
2.30686478130574781048998629109, 3.58511338042728606061281075872, 4.55273338989641640076751039960, 5.37529566414826305905694816024, 7.68868932990873702715231737160, 7.87598117685228312216412779573, 8.698370701982385061548709054403, 9.783407111288052623996500571699, 11.03981422853347132469563580569
