| L(s) = 1 | − 2.73·2-s − 7.92·3-s − 0.535·4-s − 14.8·5-s + 21.6·6-s − 3.07·7-s + 23.3·8-s + 35.8·9-s + 40.5·10-s + 11·11-s + 4.24·12-s + 8.39·14-s + 117.·15-s − 59.4·16-s − 41.2·17-s − 97.9·18-s − 139.·19-s + 7.96·20-s + 24.3·21-s − 30.0·22-s − 111.·23-s − 184.·24-s + 95.7·25-s − 70.2·27-s + 1.64·28-s − 24.9·29-s − 321.·30-s + ⋯ |
| L(s) = 1 | − 0.965·2-s − 1.52·3-s − 0.0669·4-s − 1.32·5-s + 1.47·6-s − 0.165·7-s + 1.03·8-s + 1.32·9-s + 1.28·10-s + 0.301·11-s + 0.102·12-s + 0.160·14-s + 2.02·15-s − 0.928·16-s − 0.588·17-s − 1.28·18-s − 1.68·19-s + 0.0890·20-s + 0.253·21-s − 0.291·22-s − 1.00·23-s − 1.57·24-s + 0.765·25-s − 0.500·27-s + 0.0111·28-s − 0.160·29-s − 1.95·30-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 | \( 1 + 2.73T + 8T^{2} \) |
| 3 | \( 1 + 7.92T + 27T^{2} \) |
| 5 | \( 1 + 14.8T + 125T^{2} \) |
| 7 | \( 1 + 3.07T + 343T^{2} \) |
| 17 | \( 1 + 41.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 13.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 261.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 57.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 343.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 88.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 738.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 342.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 207.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 441.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
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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
−8.384311297943294779697285171068, −7.86254375154807664653974647401, −6.85517183903682839525648121883, −6.37235020052196988075778988231, −5.17393220382415998759618632836, −4.37484778090141111958558376645, −3.81641574227619528642928317415, −1.90703160572882329407934455871, −0.59039637703878272589947666685, 0,
0.59039637703878272589947666685, 1.90703160572882329407934455871, 3.81641574227619528642928317415, 4.37484778090141111958558376645, 5.17393220382415998759618632836, 6.37235020052196988075778988231, 6.85517183903682839525648121883, 7.86254375154807664653974647401, 8.384311297943294779697285171068
