| L(s) = 1 | − 2.16·2-s − 2.96·3-s + 2.67·4-s + 2.79·5-s + 6.41·6-s + 0.0561·7-s − 1.45·8-s + 5.80·9-s − 6.04·10-s − 11-s − 7.93·12-s + 1.45·13-s − 0.121·14-s − 8.29·15-s − 2.19·16-s + 4.50·17-s − 12.5·18-s − 6.25·19-s + 7.47·20-s − 0.166·21-s + 2.16·22-s − 7.69·23-s + 4.33·24-s + 2.81·25-s − 3.13·26-s − 8.32·27-s + 0.150·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s − 1.71·3-s + 1.33·4-s + 1.24·5-s + 2.61·6-s + 0.0212·7-s − 0.516·8-s + 1.93·9-s − 1.91·10-s − 0.301·11-s − 2.29·12-s + 0.402·13-s − 0.0324·14-s − 2.14·15-s − 0.548·16-s + 1.09·17-s − 2.95·18-s − 1.43·19-s + 1.67·20-s − 0.0363·21-s + 0.460·22-s − 1.60·23-s + 0.884·24-s + 0.562·25-s − 0.615·26-s − 1.60·27-s + 0.0284·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 - 0.0561T + 7T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 23 | \( 1 + 7.69T + 23T^{2} \) |
| 29 | \( 1 + 4.54T + 29T^{2} \) |
| 31 | \( 1 - 0.500T + 31T^{2} \) |
| 37 | \( 1 - 3.93T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 - 0.495T + 43T^{2} \) |
| 47 | \( 1 - 9.84T + 47T^{2} \) |
| 53 | \( 1 + 0.774T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 3.44T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 + 0.0836T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 8.17T + 97T^{2} \) |
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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
−9.499238107715035646858561164961, −8.374717391879340878169229419312, −7.57160024265392830475580247435, −6.53778721976135723153015556292, −6.04355744039322377445634343417, −5.37545358342789733471597806023, −4.20298508798189821345896567428, −2.17234159363950747971852952350, −1.30723914005041004412703777196, 0,
1.30723914005041004412703777196, 2.17234159363950747971852952350, 4.20298508798189821345896567428, 5.37545358342789733471597806023, 6.04355744039322377445634343417, 6.53778721976135723153015556292, 7.57160024265392830475580247435, 8.374717391879340878169229419312, 9.499238107715035646858561164961
