| L(s) = 1 | + 0.839·2-s + 3-s − 1.29·4-s + 5-s + 0.839·6-s + 2.06·7-s − 2.76·8-s + 9-s + 0.839·10-s + 4.85·11-s − 1.29·12-s − 13-s + 1.72·14-s + 15-s + 0.271·16-s − 6.68·17-s + 0.839·18-s − 8.04·19-s − 1.29·20-s + 2.06·21-s + 4.07·22-s − 7.16·23-s − 2.76·24-s + 25-s − 0.839·26-s + 27-s − 2.67·28-s + ⋯ |
| L(s) = 1 | + 0.593·2-s + 0.577·3-s − 0.647·4-s + 0.447·5-s + 0.342·6-s + 0.779·7-s − 0.977·8-s + 0.333·9-s + 0.265·10-s + 1.46·11-s − 0.374·12-s − 0.277·13-s + 0.462·14-s + 0.258·15-s + 0.0677·16-s − 1.62·17-s + 0.197·18-s − 1.84·19-s − 0.289·20-s + 0.449·21-s + 0.867·22-s − 1.49·23-s − 0.564·24-s + 0.200·25-s − 0.164·26-s + 0.192·27-s − 0.504·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.839T + 2T^{2} \) |
| 7 | \( 1 - 2.06T + 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 19 | \( 1 + 8.04T + 19T^{2} \) |
| 23 | \( 1 + 7.16T + 23T^{2} \) |
| 29 | \( 1 + 4.43T + 29T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 2.14T + 43T^{2} \) |
| 47 | \( 1 + 2.65T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 + 4.38T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 - 1.44T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 + 6.71T + 97T^{2} \) |
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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
−8.000558856522734689169884497393, −6.60963197654830700732103399331, −6.48888575999811423173336810544, −5.46214119328509981791886795245, −4.48248374602064718241519500656, −4.26204884890372538647451225567, −3.47213001471005399506795057527, −2.20168748038139371251421815378, −1.70725255655970000036671300542, 0,
1.70725255655970000036671300542, 2.20168748038139371251421815378, 3.47213001471005399506795057527, 4.26204884890372538647451225567, 4.48248374602064718241519500656, 5.46214119328509981791886795245, 6.48888575999811423173336810544, 6.60963197654830700732103399331, 8.000558856522734689169884497393
