| L(s) = 1 | − 3.71·2-s + 3·3-s + 5.79·4-s − 5·5-s − 11.1·6-s − 21.2·7-s + 8.18·8-s + 9·9-s + 18.5·10-s + 17.3·12-s − 28.0·13-s + 78.7·14-s − 15·15-s − 76.7·16-s + 81.0·17-s − 33.4·18-s − 85.4·19-s − 28.9·20-s − 63.6·21-s + 145.·23-s + 24.5·24-s + 25·25-s + 104.·26-s + 27·27-s − 122.·28-s − 135.·29-s + 55.7·30-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.577·3-s + 0.724·4-s − 0.447·5-s − 0.758·6-s − 1.14·7-s + 0.361·8-s + 0.333·9-s + 0.587·10-s + 0.418·12-s − 0.597·13-s + 1.50·14-s − 0.258·15-s − 1.19·16-s + 1.15·17-s − 0.437·18-s − 1.03·19-s − 0.324·20-s − 0.661·21-s + 1.31·23-s + 0.208·24-s + 0.200·25-s + 0.784·26-s + 0.192·27-s − 0.829·28-s − 0.867·29-s + 0.339·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 3.71T + 8T^{2} \) |
| 7 | \( 1 + 21.2T + 343T^{2} \) |
| 13 | \( 1 + 28.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 81.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 34.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 105.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 540.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 682.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 604.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 32.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 278.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 325.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 529.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 675.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 492.T + 9.12e5T^{2} \) |
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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
−8.565043670958217791001951630186, −7.910424510481346598663254493121, −7.16986592658091673994494713409, −6.58878290542310411602111504015, −5.29121100638898576568426309071, −4.14868782046603218789729451379, −3.23932271738372944311283039190, −2.27766108548892149823880022280, −0.988792104905613343995472048457, 0,
0.988792104905613343995472048457, 2.27766108548892149823880022280, 3.23932271738372944311283039190, 4.14868782046603218789729451379, 5.29121100638898576568426309071, 6.58878290542310411602111504015, 7.16986592658091673994494713409, 7.910424510481346598663254493121, 8.565043670958217791001951630186
