| L(s) = 1 | + 3.98·2-s − 8.48·3-s + 7.85·4-s + 2.25·5-s − 33.7·6-s + 28.5·7-s − 0.583·8-s + 45.0·9-s + 8.99·10-s − 11·11-s − 66.6·12-s + 113.·14-s − 19.1·15-s − 65.1·16-s − 120.·17-s + 179.·18-s + 94.2·19-s + 17.7·20-s − 242.·21-s − 43.7·22-s + 92.8·23-s + 4.95·24-s − 119.·25-s − 153.·27-s + 224.·28-s + 156.·29-s − 76.3·30-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 1.63·3-s + 0.981·4-s + 0.201·5-s − 2.29·6-s + 1.54·7-s − 0.0257·8-s + 1.66·9-s + 0.284·10-s − 0.301·11-s − 1.60·12-s + 2.17·14-s − 0.329·15-s − 1.01·16-s − 1.72·17-s + 2.34·18-s + 1.13·19-s + 0.198·20-s − 2.51·21-s − 0.424·22-s + 0.841·23-s + 0.0421·24-s − 0.959·25-s − 1.09·27-s + 1.51·28-s + 1.00·29-s − 0.464·30-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 | \( 1 - 3.98T + 8T^{2} \) |
| 3 | \( 1 + 8.48T + 27T^{2} \) |
| 5 | \( 1 - 2.25T + 125T^{2} \) |
| 7 | \( 1 - 28.5T + 343T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 94.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 92.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 127.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 31.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 447.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 485.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 201.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 15.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 170.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 7.43T + 3.57e5T^{2} \) |
| 73 | \( 1 - 779.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 888.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 583.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 297.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 836.T + 9.12e5T^{2} \) |
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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
−8.368290343221256395315062855738, −7.19640596319035570031087015327, −6.65455619230892951823756454138, −5.59169319996536363739078917539, −5.28683012550316730931429803031, −4.65646366551893547534572477719, −3.94964805025962808743628104785, −2.44630244058705147524141233121, −1.37208742334019012765305436311, 0,
1.37208742334019012765305436311, 2.44630244058705147524141233121, 3.94964805025962808743628104785, 4.65646366551893547534572477719, 5.28683012550316730931429803031, 5.59169319996536363739078917539, 6.65455619230892951823756454138, 7.19640596319035570031087015327, 8.368290343221256395315062855738
