L(s) = 1 | + 0.563·2-s − 1.68·4-s − 3.19·7-s − 2.07·8-s + 2.11·11-s − 13-s − 1.80·14-s + 2.19·16-s − 4.75·17-s − 1.52·19-s + 1.19·22-s + 4.44·23-s − 0.563·26-s + 5.37·28-s + 5.63·29-s + 1.68·31-s + 5.38·32-s − 2.68·34-s + 4.20·37-s − 0.857·38-s − 0.245·41-s + 12.2·43-s − 3.56·44-s + 2.50·46-s − 9.53·47-s + 3.20·49-s + 1.68·52-s + ⋯ |
L(s) = 1 | + 0.398·2-s − 0.841·4-s − 1.20·7-s − 0.733·8-s + 0.638·11-s − 0.277·13-s − 0.481·14-s + 0.548·16-s − 1.15·17-s − 0.348·19-s + 0.254·22-s + 0.925·23-s − 0.110·26-s + 1.01·28-s + 1.04·29-s + 0.302·31-s + 0.952·32-s − 0.459·34-s + 0.691·37-s − 0.139·38-s − 0.0384·41-s + 1.87·43-s − 0.537·44-s + 0.369·46-s − 1.39·47-s + 0.457·49-s + 0.233·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 0.563T + 2T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 + 1.52T + 19T^{2} \) |
| 23 | \( 1 - 4.44T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 1.68T + 31T^{2} \) |
| 37 | \( 1 - 4.20T + 37T^{2} \) |
| 41 | \( 1 + 0.245T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 9.53T + 47T^{2} \) |
| 53 | \( 1 + 0.293T + 53T^{2} \) |
| 59 | \( 1 - 9.47T + 59T^{2} \) |
| 61 | \( 1 - 5.55T + 61T^{2} \) |
| 67 | \( 1 + 0.463T + 67T^{2} \) |
| 71 | \( 1 + 5.84T + 71T^{2} \) |
| 73 | \( 1 - 4.03T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 7.97T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.20910887100866032372435613179, −6.58978657881422210873073457491, −6.12107610064970422028723516505, −5.28069943828283924282109580758, −4.45061168270482405894479335108, −4.03121456966002013728501938518, −3.11536924774297410585927318073, −2.51682380094922928063598513868, −1.03036574031652347278664625984, 0,
1.03036574031652347278664625984, 2.51682380094922928063598513868, 3.11536924774297410585927318073, 4.03121456966002013728501938518, 4.45061168270482405894479335108, 5.28069943828283924282109580758, 6.12107610064970422028723516505, 6.58978657881422210873073457491, 7.20910887100866032372435613179