| L(s) = 1 | + 5.33·2-s + 20.4·4-s − 16.4·5-s − 9.67·7-s + 66.1·8-s − 87.4·10-s + 27.5·11-s − 51.5·14-s + 189.·16-s − 107.·17-s + 2.24·19-s − 335.·20-s + 147.·22-s − 41.8·23-s + 144.·25-s − 197.·28-s − 61.6·29-s − 191.·31-s + 480.·32-s − 575.·34-s + 158.·35-s − 98.4·37-s + 11.9·38-s − 1.08e3·40-s − 30.7·41-s + 238.·43-s + 563.·44-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.55·4-s − 1.46·5-s − 0.522·7-s + 2.92·8-s − 2.76·10-s + 0.756·11-s − 0.984·14-s + 2.95·16-s − 1.53·17-s + 0.0271·19-s − 3.74·20-s + 1.42·22-s − 0.379·23-s + 1.15·25-s − 1.33·28-s − 0.394·29-s − 1.11·31-s + 2.65·32-s − 2.90·34-s + 0.767·35-s − 0.437·37-s + 0.0511·38-s − 4.29·40-s − 0.117·41-s + 0.845·43-s + 1.92·44-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 | \( 1 - 5.33T + 8T^{2} \) |
| 5 | \( 1 + 16.4T + 125T^{2} \) |
| 7 | \( 1 + 9.67T + 343T^{2} \) |
| 11 | \( 1 - 27.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.24T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 61.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 98.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 511.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 484.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 444.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 484.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 957.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 375.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 715.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 65.5T + 9.12e5T^{2} \) |
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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
−8.453243963833539517429257371485, −7.50762985356683043206969800482, −6.81063841629145540808180810489, −6.25857306493786151174289933495, −5.12211299158593502553907042954, −4.26901241580122996639616960403, −3.79342937202612114688722291303, −3.02665424742928672294435278308, −1.80842030511212146768150094984, 0,
1.80842030511212146768150094984, 3.02665424742928672294435278308, 3.79342937202612114688722291303, 4.26901241580122996639616960403, 5.12211299158593502553907042954, 6.25857306493786151174289933495, 6.81063841629145540808180810489, 7.50762985356683043206969800482, 8.453243963833539517429257371485
