| L(s) = 1 | + 8.24·5-s + 26.8·7-s − 11·11-s − 72.1·13-s − 116.·17-s − 70.6·19-s − 21.2·23-s − 57.0·25-s − 38.4·29-s − 263.·31-s + 221.·35-s − 156.·37-s + 112.·41-s + 59.7·43-s + 134.·47-s + 376.·49-s + 585.·53-s − 90.6·55-s − 573.·59-s + 347.·61-s − 594.·65-s − 1.05e3·67-s + 292.·71-s + 230.·73-s − 295.·77-s + 1.16e3·79-s − 763.·83-s + ⋯ |
| L(s) = 1 | + 0.737·5-s + 1.44·7-s − 0.301·11-s − 1.53·13-s − 1.66·17-s − 0.852·19-s − 0.192·23-s − 0.456·25-s − 0.246·29-s − 1.52·31-s + 1.06·35-s − 0.694·37-s + 0.426·41-s + 0.211·43-s + 0.417·47-s + 1.09·49-s + 1.51·53-s − 0.222·55-s − 1.26·59-s + 0.729·61-s − 1.13·65-s − 1.92·67-s + 0.489·71-s + 0.370·73-s − 0.436·77-s + 1.66·79-s − 1.00·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 - 8.24T + 125T^{2} \) |
| 7 | \( 1 - 26.8T + 343T^{2} \) |
| 13 | \( 1 + 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 38.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 112.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 59.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 134.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 585.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 573.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 347.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.05e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 292.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 230.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 763.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 832.T + 9.12e5T^{2} \) |
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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
−9.385260194438131808730953323144, −8.692378215264208030017023417285, −7.70994655564718107662566211688, −6.97006542723097387409349922291, −5.76564217350921905409920950086, −4.95689062682147546158778242461, −4.19952040350261922975248084438, −2.34734269386630737020893559692, −1.87173455364148591309205739655, 0,
1.87173455364148591309205739655, 2.34734269386630737020893559692, 4.19952040350261922975248084438, 4.95689062682147546158778242461, 5.76564217350921905409920950086, 6.97006542723097387409349922291, 7.70994655564718107662566211688, 8.692378215264208030017023417285, 9.385260194438131808730953323144
