| L(s) = 1 | + 4.18·2-s − 8.20·3-s + 9.54·4-s + 18.8·5-s − 34.3·6-s + 14.8·7-s + 6.49·8-s + 40.3·9-s + 79.0·10-s + 11·11-s − 78.3·12-s + 62.0·14-s − 154.·15-s − 49.1·16-s − 79.2·17-s + 169.·18-s − 109.·19-s + 180.·20-s − 121.·21-s + 46.0·22-s − 112.·23-s − 53.2·24-s + 231.·25-s − 109.·27-s + 141.·28-s + 10.2·29-s − 648.·30-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 1.57·3-s + 1.19·4-s + 1.68·5-s − 2.33·6-s + 0.799·7-s + 0.286·8-s + 1.49·9-s + 2.50·10-s + 0.301·11-s − 1.88·12-s + 1.18·14-s − 2.66·15-s − 0.768·16-s − 1.13·17-s + 2.21·18-s − 1.32·19-s + 2.01·20-s − 1.26·21-s + 0.446·22-s − 1.02·23-s − 0.453·24-s + 1.84·25-s − 0.782·27-s + 0.954·28-s + 0.0657·29-s − 3.94·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.18T + 8T^{2} \) |
| 3 | \( 1 + 8.20T + 27T^{2} \) |
| 5 | \( 1 - 18.8T + 125T^{2} \) |
| 7 | \( 1 - 14.8T + 343T^{2} \) |
| 17 | \( 1 + 79.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 10.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 336.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 352.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 283.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 75.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 195.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 18.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 545.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 754.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 137.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 371.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 753.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 863.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 445.T + 9.12e5T^{2} \) |
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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
−8.591294318548652415039376641755, −6.98309114471278626042515938460, −6.40952654617959860425160961053, −5.93492190022476945603176676363, −5.20667341472953852951204460839, −4.77643054576259833138787504213, −3.85297461394952984135635607301, −2.21094505557057085163392752138, −1.69553502578698291078816266441, 0,
1.69553502578698291078816266441, 2.21094505557057085163392752138, 3.85297461394952984135635607301, 4.77643054576259833138787504213, 5.20667341472953852951204460839, 5.93492190022476945603176676363, 6.40952654617959860425160961053, 6.98309114471278626042515938460, 8.591294318548652415039376641755
