| L(s) = 1 | + 1.46·3-s − 0.928·5-s − 7.60·7-s − 24.8·9-s − 35.0·11-s + 83.5·13-s − 1.35·15-s + 17·17-s + 115.·19-s − 11.1·21-s − 64.6·23-s − 124.·25-s − 75.9·27-s + 288.·29-s + 186.·31-s − 51.2·33-s + 7.06·35-s − 241.·37-s + 122.·39-s − 43.8·41-s − 257.·43-s + 23.0·45-s − 40.1·47-s − 285.·49-s + 24.8·51-s − 214.·53-s + 32.5·55-s + ⋯ |
| L(s) = 1 | + 0.281·3-s − 0.0830·5-s − 0.410·7-s − 0.920·9-s − 0.960·11-s + 1.78·13-s − 0.0233·15-s + 0.242·17-s + 1.39·19-s − 0.115·21-s − 0.585·23-s − 0.993·25-s − 0.541·27-s + 1.84·29-s + 1.07·31-s − 0.270·33-s + 0.0341·35-s − 1.07·37-s + 0.502·39-s − 0.167·41-s − 0.913·43-s + 0.0764·45-s − 0.124·47-s − 0.831·49-s + 0.0683·51-s − 0.557·53-s + 0.0797·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 1.46T + 27T^{2} \) |
| 5 | \( 1 + 0.928T + 125T^{2} \) |
| 7 | \( 1 + 7.60T + 343T^{2} \) |
| 11 | \( 1 + 35.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 83.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 288.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 43.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 257.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 40.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 214.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 585.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 331.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 819.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 342.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 270.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 728.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 211.T + 9.12e5T^{2} \) |
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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
−8.922723032500498994338663390832, −8.276143122540459636571634658689, −7.65085262969733297581876114077, −6.34011601052226332631537881477, −5.81473190183912245583735541221, −4.74813342734349494891691899977, −3.40449090200445055138832119392, −2.93152195914184346028677183555, −1.40304645234449631957290531624, 0,
1.40304645234449631957290531624, 2.93152195914184346028677183555, 3.40449090200445055138832119392, 4.74813342734349494891691899977, 5.81473190183912245583735541221, 6.34011601052226332631537881477, 7.65085262969733297581876114077, 8.276143122540459636571634658689, 8.922723032500498994338663390832
