| L(s) = 1 | + 2·2-s − 3.34·3-s + 4·4-s − 2.71·5-s − 6.69·6-s + 8·8-s − 15.7·9-s − 5.43·10-s + 14.0·11-s − 13.3·12-s − 44.4·13-s + 9.09·15-s + 16·16-s + 107.·17-s − 31.5·18-s + 34.7·19-s − 10.8·20-s + 28.1·22-s + 23·23-s − 26.7·24-s − 117.·25-s − 88.9·26-s + 143.·27-s − 237.·29-s + 18.1·30-s + 132.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.644·3-s + 0.5·4-s − 0.242·5-s − 0.455·6-s + 0.353·8-s − 0.584·9-s − 0.171·10-s + 0.385·11-s − 0.322·12-s − 0.948·13-s + 0.156·15-s + 0.250·16-s + 1.53·17-s − 0.413·18-s + 0.419·19-s − 0.121·20-s + 0.272·22-s + 0.208·23-s − 0.227·24-s − 0.940·25-s − 0.670·26-s + 1.02·27-s − 1.52·29-s + 0.110·30-s + 0.768·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 3.34T + 27T^{2} \) |
| 5 | \( 1 + 2.71T + 125T^{2} \) |
| 11 | \( 1 - 14.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 237.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 58.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 72.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 258.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 46.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 140.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 17.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 685.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 331.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 943.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 827.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 53.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 276.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 971.T + 9.12e5T^{2} \) |
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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
−7.987321262919834974907828650195, −7.48220240716114558373000759566, −6.57376912796812470331052365630, −5.65053329984493228612841829198, −5.33986094764729337057179501080, −4.29338926312914648595229754414, −3.41675644311078445307659223894, −2.52870251468115595534059380810, −1.22200109590868485050062849448, 0,
1.22200109590868485050062849448, 2.52870251468115595534059380810, 3.41675644311078445307659223894, 4.29338926312914648595229754414, 5.33986094764729337057179501080, 5.65053329984493228612841829198, 6.57376912796812470331052365630, 7.48220240716114558373000759566, 7.987321262919834974907828650195
