| L(s) = 1 | − 1.65·2-s + 1.58·3-s + 0.752·4-s + 0.552·5-s − 2.62·6-s − 2.50·7-s + 2.06·8-s − 0.492·9-s − 0.916·10-s + 2.83·11-s + 1.19·12-s − 4.50·13-s + 4.16·14-s + 0.874·15-s − 4.93·16-s + 4.73·17-s + 0.816·18-s − 1.20·19-s + 0.415·20-s − 3.97·21-s − 4.71·22-s + 1.59·23-s + 3.27·24-s − 4.69·25-s + 7.47·26-s − 5.53·27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.914·3-s + 0.376·4-s + 0.247·5-s − 1.07·6-s − 0.947·7-s + 0.731·8-s − 0.164·9-s − 0.289·10-s + 0.856·11-s + 0.344·12-s − 1.24·13-s + 1.11·14-s + 0.225·15-s − 1.23·16-s + 1.14·17-s + 0.192·18-s − 0.276·19-s + 0.0929·20-s − 0.866·21-s − 1.00·22-s + 0.333·23-s + 0.668·24-s − 0.938·25-s + 1.46·26-s − 1.06·27-s − 0.356·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2011 | \( 1 + T \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 - 0.552T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.83T + 11T^{2} \) |
| 13 | \( 1 + 4.50T + 13T^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 - 1.59T + 23T^{2} \) |
| 29 | \( 1 - 6.20T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 + 7.93T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 + 0.500T + 43T^{2} \) |
| 47 | \( 1 - 1.74T + 47T^{2} \) |
| 53 | \( 1 - 2.10T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 - 6.87T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 1.51T + 79T^{2} \) |
| 83 | \( 1 + 2.86T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 - 9.99T + 97T^{2} \) |
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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
−9.062470690744318804516139562803, −8.090657786218644929562886537215, −7.50909109158481237874790168752, −6.72235426035537342904636752749, −5.74283308727230925351350751458, −4.56008255872072033772778290535, −3.48027743500619368220184384046, −2.63439414669025159728034959627, −1.51766661434460875740606317354, 0,
1.51766661434460875740606317354, 2.63439414669025159728034959627, 3.48027743500619368220184384046, 4.56008255872072033772778290535, 5.74283308727230925351350751458, 6.72235426035537342904636752749, 7.50909109158481237874790168752, 8.090657786218644929562886537215, 9.062470690744318804516139562803
