L(s) = 1 | + 2.14·2-s + 2.59·4-s + 5-s − 1.17·7-s + 1.27·8-s + 2.14·10-s − 4.28·11-s − 3.87·13-s − 2.51·14-s − 2.45·16-s + 2.72·17-s − 3.65·19-s + 2.59·20-s − 9.19·22-s − 1.11·23-s + 25-s − 8.30·26-s − 3.04·28-s + 0.396·29-s + 4.84·31-s − 7.81·32-s + 5.83·34-s − 1.17·35-s − 4.02·37-s − 7.82·38-s + 1.27·40-s + 1.59·41-s + ⋯ |
L(s) = 1 | + 1.51·2-s + 1.29·4-s + 0.447·5-s − 0.443·7-s + 0.451·8-s + 0.677·10-s − 1.29·11-s − 1.07·13-s − 0.672·14-s − 0.613·16-s + 0.660·17-s − 0.837·19-s + 0.580·20-s − 1.95·22-s − 0.233·23-s + 0.200·25-s − 1.62·26-s − 0.576·28-s + 0.0736·29-s + 0.869·31-s − 1.38·32-s + 1.00·34-s − 0.198·35-s − 0.661·37-s − 1.26·38-s + 0.202·40-s + 0.249·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 7 | \( 1 + 1.17T + 7T^{2} \) |
| 11 | \( 1 + 4.28T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 - 0.396T + 29T^{2} \) |
| 31 | \( 1 - 4.84T + 31T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 - 1.59T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 6.65T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 0.356T + 59T^{2} \) |
| 61 | \( 1 - 7.28T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 2.55T + 71T^{2} \) |
| 73 | \( 1 + 0.593T + 73T^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 97 | \( 1 - 0.629T + 97T^{2} \) |
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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
−7.895791456100118669683889689045, −7.08149543199451856071353494838, −6.36147087755274400199800920093, −5.69333514950574571630780356292, −5.00007536439058911221619875776, −4.50395435522721923500352104959, −3.35405572993031034727139376525, −2.77124037225154948895341335245, −1.96999653325533037337038288645, 0,
1.96999653325533037337038288645, 2.77124037225154948895341335245, 3.35405572993031034727139376525, 4.50395435522721923500352104959, 5.00007536439058911221619875776, 5.69333514950574571630780356292, 6.36147087755274400199800920093, 7.08149543199451856071353494838, 7.895791456100118669683889689045