| L(s) = 1 | − 1.37·2-s − 2.08·3-s − 0.110·4-s + 2.34·5-s + 2.86·6-s − 1.29·7-s + 2.90·8-s + 1.34·9-s − 3.22·10-s + 11-s + 0.230·12-s + 1.78·14-s − 4.88·15-s − 3.76·16-s − 5.06·17-s − 1.85·18-s + 1.52·19-s − 0.258·20-s + 2.70·21-s − 1.37·22-s + 3.16·23-s − 6.04·24-s + 0.489·25-s + 3.44·27-s + 0.143·28-s − 4.07·29-s + 6.71·30-s + ⋯ |
| L(s) = 1 | − 0.971·2-s − 1.20·3-s − 0.0552·4-s + 1.04·5-s + 1.17·6-s − 0.489·7-s + 1.02·8-s + 0.448·9-s − 1.01·10-s + 0.301·11-s + 0.0664·12-s + 0.476·14-s − 1.26·15-s − 0.941·16-s − 1.22·17-s − 0.436·18-s + 0.349·19-s − 0.0578·20-s + 0.589·21-s − 0.293·22-s + 0.659·23-s − 1.23·24-s + 0.0978·25-s + 0.663·27-s + 0.0270·28-s − 0.757·29-s + 1.22·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.37T + 2T^{2} \) |
| 3 | \( 1 + 2.08T + 3T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 7 | \( 1 + 1.29T + 7T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 - 1.52T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + 3.69T + 31T^{2} \) |
| 37 | \( 1 - 0.984T + 37T^{2} \) |
| 41 | \( 1 - 4.53T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.71T + 47T^{2} \) |
| 53 | \( 1 - 7.08T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 9.99T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 7.43T + 73T^{2} \) |
| 79 | \( 1 - 3.76T + 79T^{2} \) |
| 83 | \( 1 + 2.79T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 8.07T + 97T^{2} \) |
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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
−9.258254084684838660078587712902, −8.222244411432324770044941914245, −7.11394567811052458674430054924, −6.51450633223880470014905484147, −5.69273637095872405901752809888, −5.02237944156835460926286783190, −3.99855742888996789339711002280, −2.39708730703985393621833098546, −1.22447672538704992756243919704, 0,
1.22447672538704992756243919704, 2.39708730703985393621833098546, 3.99855742888996789339711002280, 5.02237944156835460926286783190, 5.69273637095872405901752809888, 6.51450633223880470014905484147, 7.11394567811052458674430054924, 8.222244411432324770044941914245, 9.258254084684838660078587712902
