| L(s) = 1 | − 0.564·3-s − 4.10·5-s − 4.97·7-s − 2.68·9-s − 0.0769·11-s − 3.33·13-s + 2.31·15-s − 5.96·17-s + 7.28·19-s + 2.80·21-s − 2.10·23-s + 11.8·25-s + 3.20·27-s + 3.24·29-s + 6.20·31-s + 0.0434·33-s + 20.3·35-s + 2.16·37-s + 1.88·39-s − 5.00·41-s + 1.06·43-s + 11.0·45-s − 5.86·47-s + 17.7·49-s + 3.36·51-s − 9.01·53-s + 0.315·55-s + ⋯ |
| L(s) = 1 | − 0.325·3-s − 1.83·5-s − 1.87·7-s − 0.893·9-s − 0.0232·11-s − 0.923·13-s + 0.598·15-s − 1.44·17-s + 1.67·19-s + 0.612·21-s − 0.439·23-s + 2.36·25-s + 0.617·27-s + 0.601·29-s + 1.11·31-s + 0.00756·33-s + 3.44·35-s + 0.355·37-s + 0.301·39-s − 0.781·41-s + 0.162·43-s + 1.63·45-s − 0.855·47-s + 2.53·49-s + 0.471·51-s − 1.23·53-s + 0.0425·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 + 0.564T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 11 | \( 1 + 0.0769T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 5.96T + 17T^{2} \) |
| 19 | \( 1 - 7.28T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 - 6.20T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 - 1.06T + 43T^{2} \) |
| 47 | \( 1 + 5.86T + 47T^{2} \) |
| 53 | \( 1 + 9.01T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 4.58T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 3.35T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 - 2.00T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 + 8.84T + 89T^{2} \) |
| 97 | \( 1 + 0.394T + 97T^{2} \) |
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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
−7.37833715729988654727591171524, −6.67355604008954646111428633254, −6.33556572288279932894982760648, −5.22953448617311024527794725580, −4.58687883429270254263604266921, −3.74596534248004570627176204600, −3.09645751257097728903306149890, −2.64980048862778514037263762424, −0.64992242506808053110890548958, 0,
0.64992242506808053110890548958, 2.64980048862778514037263762424, 3.09645751257097728903306149890, 3.74596534248004570627176204600, 4.58687883429270254263604266921, 5.22953448617311024527794725580, 6.33556572288279932894982760648, 6.67355604008954646111428633254, 7.37833715729988654727591171524
