| L(s) = 1 | + 4.27·2-s − 7.17·3-s + 10.2·4-s − 5·5-s − 30.6·6-s + 5.20·7-s + 9.78·8-s + 24.4·9-s − 21.3·10-s + 46.2·11-s − 73.7·12-s + 22.2·14-s + 35.8·15-s − 40.4·16-s + 14.9·17-s + 104.·18-s − 75.6·19-s − 51.4·20-s − 37.3·21-s + 197.·22-s + 134.·23-s − 70.1·24-s + 25·25-s + 18.2·27-s + 53.5·28-s − 236.·29-s + 153.·30-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 1.38·3-s + 1.28·4-s − 0.447·5-s − 2.08·6-s + 0.281·7-s + 0.432·8-s + 0.905·9-s − 0.676·10-s + 1.26·11-s − 1.77·12-s + 0.425·14-s + 0.617·15-s − 0.632·16-s + 0.212·17-s + 1.36·18-s − 0.914·19-s − 0.575·20-s − 0.388·21-s + 1.91·22-s + 1.22·23-s − 0.596·24-s + 0.200·25-s + 0.130·27-s + 0.361·28-s − 1.51·29-s + 0.933·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.27T + 8T^{2} \) |
| 3 | \( 1 + 7.17T + 27T^{2} \) |
| 7 | \( 1 - 5.20T + 343T^{2} \) |
| 11 | \( 1 - 46.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 14.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 75.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 65.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 311.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 86.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 108.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 700.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 287.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 59.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 304.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 274.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 835.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 382.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.49e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3T + 9.12e5T^{2} \) |
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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
−9.527305584159843397357051268026, −8.462781982379936451975343687616, −6.99513955938993475819644892335, −6.56106809770528012448405488904, −5.65578068547923085093176807277, −4.92433513643470122754518588365, −4.18561092634141280017893749586, −3.25889635565543021201752253144, −1.56254481572502820800695790331, 0,
1.56254481572502820800695790331, 3.25889635565543021201752253144, 4.18561092634141280017893749586, 4.92433513643470122754518588365, 5.65578068547923085093176807277, 6.56106809770528012448405488904, 6.99513955938993475819644892335, 8.462781982379936451975343687616, 9.527305584159843397357051268026
