| L(s) = 1 | + 0.919·2-s − 7.15·4-s − 26.1·7-s − 13.9·8-s − 11·11-s + 46.5·13-s − 24.0·14-s + 44.4·16-s − 11.5·17-s + 68.8·19-s − 10.1·22-s − 30.7·23-s + 42.8·26-s + 186.·28-s − 37.1·29-s − 144.·31-s + 152.·32-s − 10.6·34-s − 55.6·37-s + 63.3·38-s − 239.·41-s + 142.·43-s + 78.7·44-s − 28.2·46-s + 561.·47-s + 339.·49-s − 333.·52-s + ⋯ |
| L(s) = 1 | + 0.324·2-s − 0.894·4-s − 1.41·7-s − 0.615·8-s − 0.301·11-s + 0.993·13-s − 0.458·14-s + 0.694·16-s − 0.164·17-s + 0.831·19-s − 0.0979·22-s − 0.278·23-s + 0.322·26-s + 1.26·28-s − 0.237·29-s − 0.834·31-s + 0.841·32-s − 0.0535·34-s − 0.247·37-s + 0.270·38-s − 0.913·41-s + 0.505·43-s + 0.269·44-s − 0.0905·46-s + 1.74·47-s + 0.990·49-s − 0.888·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 0.919T + 8T^{2} \) |
| 7 | \( 1 + 26.1T + 343T^{2} \) |
| 13 | \( 1 - 46.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 68.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 37.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 55.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 239.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 561.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 48.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 719.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 511.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 438.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.07e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 952.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 710.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 563.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 766.T + 9.12e5T^{2} \) |
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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
−8.384331997170548784024090119945, −7.36841496698873265734539900369, −6.52774224102115024207260197546, −5.74773063812808550072333316884, −5.17583400646250347435891506580, −3.86313901294354946778886927346, −3.60602200925215323855332137779, −2.54316480683334890572911519004, −0.971326939047106850310856505433, 0,
0.971326939047106850310856505433, 2.54316480683334890572911519004, 3.60602200925215323855332137779, 3.86313901294354946778886927346, 5.17583400646250347435891506580, 5.74773063812808550072333316884, 6.52774224102115024207260197546, 7.36841496698873265734539900369, 8.384331997170548784024090119945
