| L(s) = 1 | − 5.14·2-s − 6.70·3-s + 18.4·4-s − 17.4·5-s + 34.4·6-s + 3.29·7-s − 53.8·8-s + 17.9·9-s + 89.9·10-s − 123.·12-s − 38.6·13-s − 16.9·14-s + 117.·15-s + 129.·16-s − 17·17-s − 92.2·18-s − 73.9·19-s − 323.·20-s − 22.0·21-s + 50.9·23-s + 360.·24-s + 181.·25-s + 198.·26-s + 60.8·27-s + 60.7·28-s + 69.3·29-s − 603.·30-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 1.28·3-s + 2.30·4-s − 1.56·5-s + 2.34·6-s + 0.177·7-s − 2.37·8-s + 0.663·9-s + 2.84·10-s − 2.97·12-s − 0.824·13-s − 0.323·14-s + 2.01·15-s + 2.01·16-s − 0.242·17-s − 1.20·18-s − 0.893·19-s − 3.61·20-s − 0.229·21-s + 0.461·23-s + 3.06·24-s + 1.44·25-s + 1.50·26-s + 0.433·27-s + 0.410·28-s + 0.444·29-s − 3.67·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 5.14T + 8T^{2} \) |
| 3 | \( 1 + 6.70T + 27T^{2} \) |
| 5 | \( 1 + 17.4T + 125T^{2} \) |
| 7 | \( 1 - 3.29T + 343T^{2} \) |
| 13 | \( 1 + 38.6T + 2.19e3T^{2} \) |
| 19 | \( 1 + 73.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 154.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 112.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 312.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 654.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 861.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 432.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 374.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 915.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 158.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.09e3T + 9.12e5T^{2} \) |
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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
−8.420698303081194791685686110939, −7.64857699736084138486790655019, −7.03392582454406467897055446601, −6.48172743857995261273640355649, −5.30359142470593422848941573530, −4.40147273278084591650797420069, −3.14930023264706618901076800633, −1.82658927863249627124333172994, −0.57727140084173163847048911105, 0,
0.57727140084173163847048911105, 1.82658927863249627124333172994, 3.14930023264706618901076800633, 4.40147273278084591650797420069, 5.30359142470593422848941573530, 6.48172743857995261273640355649, 7.03392582454406467897055446601, 7.64857699736084138486790655019, 8.420698303081194791685686110939
