| L(s) = 1 | − 4.89·2-s + 3·3-s + 15.9·4-s − 7.63·5-s − 14.6·6-s + 26.7·7-s − 39.1·8-s + 9·9-s + 37.3·10-s + 38.7·11-s + 47.9·12-s − 67.7·13-s − 130.·14-s − 22.8·15-s + 63.7·16-s + 64.1·17-s − 44.0·18-s − 11.3·19-s − 122.·20-s + 80.1·21-s − 189.·22-s + 23·23-s − 117.·24-s − 66.7·25-s + 331.·26-s + 27·27-s + 427.·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.577·3-s + 1.99·4-s − 0.682·5-s − 0.999·6-s + 1.44·7-s − 1.72·8-s + 0.333·9-s + 1.18·10-s + 1.06·11-s + 1.15·12-s − 1.44·13-s − 2.49·14-s − 0.394·15-s + 0.996·16-s + 0.915·17-s − 0.577·18-s − 0.137·19-s − 1.36·20-s + 0.832·21-s − 1.84·22-s + 0.208·23-s − 0.998·24-s − 0.534·25-s + 2.50·26-s + 0.192·27-s + 2.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 4.89T + 8T^{2} \) |
| 5 | \( 1 + 7.63T + 125T^{2} \) |
| 7 | \( 1 - 26.7T + 343T^{2} \) |
| 11 | \( 1 - 38.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.3T + 6.85e3T^{2} \) |
| 31 | \( 1 - 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 393.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 114.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 274.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 552.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 535.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 372.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 148.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 4.05T + 3.89e5T^{2} \) |
| 79 | \( 1 + 788.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 255.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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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
−8.482745708492938030195681568018, −7.72205616860767031498695588054, −7.42926250770087926060406685942, −6.55548349466791326207383001241, −5.12969273530395046760341966994, −4.24221381712200397654526345964, −3.02832187358006029840348778659, −1.88927907198376315110515291198, −1.27063952265120053454133789274, 0,
1.27063952265120053454133789274, 1.88927907198376315110515291198, 3.02832187358006029840348778659, 4.24221381712200397654526345964, 5.12969273530395046760341966994, 6.55548349466791326207383001241, 7.42926250770087926060406685942, 7.72205616860767031498695588054, 8.482745708492938030195681568018
