L(s) = 1 | + 0.694·2-s − 1.51·4-s + 5-s − 0.291·7-s − 2.44·8-s + 0.694·10-s − 1.78·11-s + 2.19·13-s − 0.202·14-s + 1.33·16-s + 5.46·17-s − 6.84·19-s − 1.51·20-s − 1.24·22-s − 5.34·23-s + 25-s + 1.52·26-s + 0.442·28-s + 2.95·29-s + 6.22·31-s + 5.81·32-s + 3.79·34-s − 0.291·35-s − 8.54·37-s − 4.75·38-s − 2.44·40-s − 4.55·41-s + ⋯ |
L(s) = 1 | + 0.490·2-s − 0.758·4-s + 0.447·5-s − 0.110·7-s − 0.863·8-s + 0.219·10-s − 0.539·11-s + 0.607·13-s − 0.0541·14-s + 0.334·16-s + 1.32·17-s − 1.57·19-s − 0.339·20-s − 0.264·22-s − 1.11·23-s + 0.200·25-s + 0.298·26-s + 0.0836·28-s + 0.548·29-s + 1.11·31-s + 1.02·32-s + 0.650·34-s − 0.0492·35-s − 1.40·37-s − 0.771·38-s − 0.386·40-s − 0.711·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 2 | \( 1 - 0.694T + 2T^{2} \) |
| 7 | \( 1 + 0.291T + 7T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 - 2.95T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 + 8.54T + 37T^{2} \) |
| 41 | \( 1 + 4.55T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 0.296T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 - 6.21T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−8.390219175440043346610377256623, −7.51283025020522425080777147015, −6.26790704701353693798105016438, −6.00063735660804244733089325810, −5.07827310728451137311855990130, −4.38630876284647401551644582326, −3.54075605339227292833957545387, −2.71956859974510027429541606030, −1.46141341479169766771332739064, 0,
1.46141341479169766771332739064, 2.71956859974510027429541606030, 3.54075605339227292833957545387, 4.38630876284647401551644582326, 5.07827310728451137311855990130, 6.00063735660804244733089325810, 6.26790704701353693798105016438, 7.51283025020522425080777147015, 8.390219175440043346610377256623