| L(s) = 1 | − 5.32·3-s − 18.8·5-s − 7·7-s + 1.39·9-s + 7.39·11-s − 79.7·13-s + 100.·15-s + 73.0·17-s + 142.·19-s + 37.3·21-s − 15.8·23-s + 229.·25-s + 136.·27-s − 162.·29-s + 333.·31-s − 39.4·33-s + 131.·35-s − 371.·37-s + 425.·39-s − 41·41-s − 78.0·43-s − 26.2·45-s + 169.·47-s + 49·49-s − 389.·51-s + 748.·53-s − 139.·55-s + ⋯ |
| L(s) = 1 | − 1.02·3-s − 1.68·5-s − 0.377·7-s + 0.0516·9-s + 0.202·11-s − 1.70·13-s + 1.72·15-s + 1.04·17-s + 1.71·19-s + 0.387·21-s − 0.143·23-s + 1.83·25-s + 0.972·27-s − 1.03·29-s + 1.93·31-s − 0.207·33-s + 0.636·35-s − 1.64·37-s + 1.74·39-s − 0.156·41-s − 0.276·43-s − 0.0870·45-s + 0.527·47-s + 0.142·49-s − 1.06·51-s + 1.94·53-s − 0.341·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 + 5.32T + 27T^{2} \) |
| 5 | \( 1 + 18.8T + 125T^{2} \) |
| 11 | \( 1 - 7.39T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 73.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 333.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 371.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 78.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 748.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 696.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 192.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 48.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 359.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 473.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 743.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 976.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−9.052648688563677769023385331435, −7.86088773956039768917477800123, −7.44069480510203203125932146696, −6.61582179149171897564699934264, −5.38740338235138938899837147625, −4.86766932526938445460614447549, −3.71822401381469465488450820287, −2.88360165452041423492121490750, −0.865677055076055684640704879300, 0,
0.865677055076055684640704879300, 2.88360165452041423492121490750, 3.71822401381469465488450820287, 4.86766932526938445460614447549, 5.38740338235138938899837147625, 6.61582179149171897564699934264, 7.44069480510203203125932146696, 7.86088773956039768917477800123, 9.052648688563677769023385331435
