| L(s) = 1 | + 3·3-s + 5·5-s − 14.8·7-s + 9·9-s − 49.1·11-s + 13·13-s + 15·15-s − 27.0·17-s + 13.6·19-s − 44.5·21-s + 172.·23-s + 25·25-s + 27·27-s + 199.·29-s − 5.65·31-s − 147.·33-s − 74.2·35-s + 356.·37-s + 39·39-s − 219.·41-s − 542.·43-s + 45·45-s − 342.·47-s − 122.·49-s − 81.2·51-s − 355.·53-s − 245.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.801·7-s + 0.333·9-s − 1.34·11-s + 0.277·13-s + 0.258·15-s − 0.386·17-s + 0.164·19-s − 0.462·21-s + 1.56·23-s + 0.200·25-s + 0.192·27-s + 1.27·29-s − 0.0327·31-s − 0.777·33-s − 0.358·35-s + 1.58·37-s + 0.160·39-s − 0.837·41-s − 1.92·43-s + 0.149·45-s − 1.06·47-s − 0.357·49-s − 0.222·51-s − 0.921·53-s − 0.601·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 13 | \( 1 - 13T \) |
| good | 7 | \( 1 + 14.8T + 343T^{2} \) |
| 11 | \( 1 + 49.1T + 1.33e3T^{2} \) |
| 17 | \( 1 + 27.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 199.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 5.65T + 2.97e4T^{2} \) |
| 37 | \( 1 - 356.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 219.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 542.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 355.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 134.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 188.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 309.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 73.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 696.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 877.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 984.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 606.T + 9.12e5T^{2} \) |
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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
−8.631817966849149392824575614314, −8.028498245499090363385533054948, −6.99403097410927963047837807668, −6.37806477762911473147331114695, −5.30188548077868048681934551547, −4.53740138147714388866472792721, −3.11054341301676490799985330037, −2.78468347581808261602305180278, −1.42350135201902743962023027999, 0,
1.42350135201902743962023027999, 2.78468347581808261602305180278, 3.11054341301676490799985330037, 4.53740138147714388866472792721, 5.30188548077868048681934551547, 6.37806477762911473147331114695, 6.99403097410927963047837807668, 8.028498245499090363385533054948, 8.631817966849149392824575614314
