| L(s) = 1 | − 3.12·2-s − 0.537·3-s + 1.77·4-s − 0.516·5-s + 1.68·6-s − 34.6·7-s + 19.4·8-s − 26.7·9-s + 1.61·10-s + 11·11-s − 0.955·12-s + 108.·14-s + 0.277·15-s − 75.0·16-s + 114.·17-s + 83.5·18-s − 122.·19-s − 0.917·20-s + 18.6·21-s − 34.3·22-s − 66.8·23-s − 10.4·24-s − 124.·25-s + 28.8·27-s − 61.5·28-s + 64.6·29-s − 0.867·30-s + ⋯ |
| L(s) = 1 | − 1.10·2-s − 0.103·3-s + 0.222·4-s − 0.0461·5-s + 0.114·6-s − 1.87·7-s + 0.859·8-s − 0.989·9-s + 0.0510·10-s + 0.301·11-s − 0.0229·12-s + 2.06·14-s + 0.00477·15-s − 1.17·16-s + 1.62·17-s + 1.09·18-s − 1.48·19-s − 0.0102·20-s + 0.193·21-s − 0.333·22-s − 0.606·23-s − 0.0889·24-s − 0.997·25-s + 0.205·27-s − 0.415·28-s + 0.414·29-s − 0.00528·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.12T + 8T^{2} \) |
| 3 | \( 1 + 0.537T + 27T^{2} \) |
| 5 | \( 1 + 0.516T + 125T^{2} \) |
| 7 | \( 1 + 34.6T + 343T^{2} \) |
| 17 | \( 1 - 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 64.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 238.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 28.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 308.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 246.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 165.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 79.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 267.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 351.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 615.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 470.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 65.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 982.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 448.T + 9.12e5T^{2} \) |
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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
−8.468584103946863924311617846688, −8.051236191889409926114174716367, −6.90533620469871084374282187777, −6.28527789775090590127380392602, −5.52297957764792452717124025870, −4.15303338561771109404131177728, −3.32156619766062901941907639122, −2.29594541170171990766312961935, −0.794127741725024685317978883392, 0,
0.794127741725024685317978883392, 2.29594541170171990766312961935, 3.32156619766062901941907639122, 4.15303338561771109404131177728, 5.52297957764792452717124025870, 6.28527789775090590127380392602, 6.90533620469871084374282187777, 8.051236191889409926114174716367, 8.468584103946863924311617846688
