| L(s) = 1 | − 2-s + 0.669·3-s + 4-s + 2.50·5-s − 0.669·6-s − 4.17·7-s − 8-s − 2.55·9-s − 2.50·10-s − 11-s + 0.669·12-s + 4.30·13-s + 4.17·14-s + 1.68·15-s + 16-s − 7.27·17-s + 2.55·18-s − 5.88·19-s + 2.50·20-s − 2.79·21-s + 22-s − 4.05·23-s − 0.669·24-s + 1.29·25-s − 4.30·26-s − 3.71·27-s − 4.17·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.386·3-s + 0.5·4-s + 1.12·5-s − 0.273·6-s − 1.57·7-s − 0.353·8-s − 0.850·9-s − 0.793·10-s − 0.301·11-s + 0.193·12-s + 1.19·13-s + 1.11·14-s + 0.433·15-s + 0.250·16-s − 1.76·17-s + 0.601·18-s − 1.35·19-s + 0.560·20-s − 0.610·21-s + 0.213·22-s − 0.845·23-s − 0.136·24-s + 0.258·25-s − 0.843·26-s − 0.715·27-s − 0.789·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 0.669T + 3T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 - 7.98T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 5.71T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 9.11T + 67T^{2} \) |
| 71 | \( 1 + 2.03T + 71T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 9.40T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−9.496098519278261635490930706984, −8.802218191040943316739768228215, −8.426756729019950887095883410944, −6.84947802787363834408255838710, −6.30568317123496209721815060897, −5.72044533622396695164856592089, −3.98369300716670292863683894867, −2.81765266188659653155758160744, −2.04054599148389833602769144787, 0,
2.04054599148389833602769144787, 2.81765266188659653155758160744, 3.98369300716670292863683894867, 5.72044533622396695164856592089, 6.30568317123496209721815060897, 6.84947802787363834408255838710, 8.426756729019950887095883410944, 8.802218191040943316739768228215, 9.496098519278261635490930706984
