| L(s) = 1 | − 1.61·2-s − 5.39·4-s − 1.20·5-s + 28.2·7-s + 21.6·8-s + 1.95·10-s + 31.7·11-s − 45.6·14-s + 8.22·16-s + 16.0·17-s − 58.5·19-s + 6.51·20-s − 51.3·22-s − 152.·23-s − 123.·25-s − 152.·28-s − 265.·29-s − 56.9·31-s − 186.·32-s − 25.9·34-s − 34.1·35-s + 444.·37-s + 94.5·38-s − 26.1·40-s + 189.·41-s − 132.·43-s − 171.·44-s + ⋯ |
| L(s) = 1 | − 0.570·2-s − 0.674·4-s − 0.108·5-s + 1.52·7-s + 0.955·8-s + 0.0617·10-s + 0.871·11-s − 0.871·14-s + 0.128·16-s + 0.229·17-s − 0.706·19-s + 0.0728·20-s − 0.497·22-s − 1.38·23-s − 0.988·25-s − 1.02·28-s − 1.69·29-s − 0.329·31-s − 1.02·32-s − 0.131·34-s − 0.165·35-s + 1.97·37-s + 0.403·38-s − 0.103·40-s + 0.720·41-s − 0.470·43-s − 0.587·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 8T^{2} \) |
| 5 | \( 1 + 1.20T + 125T^{2} \) |
| 7 | \( 1 - 28.2T + 343T^{2} \) |
| 11 | \( 1 - 31.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 16.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 56.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 444.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 189.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 513.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 619.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 597.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 332.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 679.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 88.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 154.T + 9.12e5T^{2} \) |
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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
−8.671827562169545836659344172138, −7.88515946981076392102220930007, −7.60303358017391370686831235784, −6.19503484286814273970397197542, −5.33485086158298960138531368502, −4.30883757976928540130942660969, −3.91759483809637335422971975779, −2.04809871847129013814105246116, −1.31268941788081637290527637344, 0,
1.31268941788081637290527637344, 2.04809871847129013814105246116, 3.91759483809637335422971975779, 4.30883757976928540130942660969, 5.33485086158298960138531368502, 6.19503484286814273970397197542, 7.60303358017391370686831235784, 7.88515946981076392102220930007, 8.671827562169545836659344172138
