| L(s) = 1 | − 3.54·2-s − 3.02·3-s + 4.56·4-s + 20.5·5-s + 10.7·6-s + 15.8·7-s + 12.1·8-s − 17.8·9-s − 72.9·10-s − 34.9·11-s − 13.8·12-s − 56.8·13-s − 56.1·14-s − 62.3·15-s − 79.6·16-s + 107.·17-s + 63.1·18-s + 82.7·19-s + 93.9·20-s − 47.9·21-s + 123.·22-s − 34.1·23-s − 36.9·24-s + 299.·25-s + 201.·26-s + 135.·27-s + 72.2·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.582·3-s + 0.570·4-s + 1.84·5-s + 0.730·6-s + 0.855·7-s + 0.538·8-s − 0.660·9-s − 2.30·10-s − 0.957·11-s − 0.332·12-s − 1.21·13-s − 1.07·14-s − 1.07·15-s − 1.24·16-s + 1.53·17-s + 0.827·18-s + 0.998·19-s + 1.05·20-s − 0.498·21-s + 1.19·22-s − 0.309·23-s − 0.313·24-s + 2.39·25-s + 1.51·26-s + 0.967·27-s + 0.487·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 3.54T + 8T^{2} \) |
| 3 | \( 1 + 3.02T + 27T^{2} \) |
| 5 | \( 1 - 20.5T + 125T^{2} \) |
| 7 | \( 1 - 15.8T + 343T^{2} \) |
| 11 | \( 1 + 34.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 34.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 201.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 43.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 170.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 28.1T + 6.89e4T^{2} \) |
| 47 | \( 1 - 37.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 249.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 8.65T + 2.05e5T^{2} \) |
| 61 | \( 1 + 133.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 660.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 573.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 403.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 104.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 194.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
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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
−8.616033455277269270467414564686, −7.76111071561790006645049928709, −7.20659275611438542179203809399, −5.94185306866102635050543899607, −5.31270777039523502679930625547, −4.94985489371774095281155353853, −2.92474938997300243861610376272, −1.99049466896823500226898975792, −1.21673122522232478758474748754, 0,
1.21673122522232478758474748754, 1.99049466896823500226898975792, 2.92474938997300243861610376272, 4.94985489371774095281155353853, 5.31270777039523502679930625547, 5.94185306866102635050543899607, 7.20659275611438542179203809399, 7.76111071561790006645049928709, 8.616033455277269270467414564686
