| L(s) = 1 | − 2.15·2-s + 9.55·3-s − 3.37·4-s − 17.5·5-s − 20.5·6-s + 3.15·7-s + 24.4·8-s + 64.2·9-s + 37.6·10-s − 12.8·11-s − 32.2·12-s + 2.45·13-s − 6.77·14-s − 167.·15-s − 25.5·16-s − 62.6·17-s − 138.·18-s + 39.6·19-s + 59.1·20-s + 30.1·21-s + 27.6·22-s + 233.·24-s + 181.·25-s − 5.27·26-s + 355.·27-s − 10.6·28-s − 187.·29-s + ⋯ |
| L(s) = 1 | − 0.760·2-s + 1.83·3-s − 0.422·4-s − 1.56·5-s − 1.39·6-s + 0.170·7-s + 1.08·8-s + 2.37·9-s + 1.19·10-s − 0.353·11-s − 0.775·12-s + 0.0523·13-s − 0.129·14-s − 2.87·15-s − 0.399·16-s − 0.893·17-s − 1.80·18-s + 0.478·19-s + 0.660·20-s + 0.312·21-s + 0.268·22-s + 1.98·24-s + 1.45·25-s − 0.0398·26-s + 2.53·27-s − 0.0718·28-s − 1.20·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 + 2.15T + 8T^{2} \) |
| 3 | \( 1 - 9.55T + 27T^{2} \) |
| 5 | \( 1 + 17.5T + 125T^{2} \) |
| 7 | \( 1 - 3.15T + 343T^{2} \) |
| 11 | \( 1 + 12.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.45T + 2.19e3T^{2} \) |
| 17 | \( 1 + 62.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 3.14T + 2.97e4T^{2} \) |
| 37 | \( 1 + 132.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 235.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 128.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 189.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 503.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 48.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 490.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 994.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 235.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−9.635968402708893865963994524367, −8.875114663131723560200193718589, −8.292754441692587498338701641513, −7.68746277743436316037647896364, −7.10703762661002619051195209821, −4.75086124143907554746082359576, −3.98295339999051901044090089311, −3.12496555524837484428391818739, −1.64275437745992140349219718009, 0,
1.64275437745992140349219718009, 3.12496555524837484428391818739, 3.98295339999051901044090089311, 4.75086124143907554746082359576, 7.10703762661002619051195209821, 7.68746277743436316037647896364, 8.292754441692587498338701641513, 8.875114663131723560200193718589, 9.635968402708893865963994524367
