| L(s) = 1 | + 1.49·2-s − 8.91·3-s − 5.77·4-s + 12.8·5-s − 13.2·6-s − 33.8·7-s − 20.5·8-s + 52.4·9-s + 19.2·10-s − 11·11-s + 51.5·12-s − 50.5·14-s − 114.·15-s + 15.6·16-s − 129.·17-s + 78.1·18-s + 74.1·19-s − 74.5·20-s + 302.·21-s − 16.3·22-s + 167.·23-s + 183.·24-s + 41.1·25-s − 227.·27-s + 195.·28-s − 95.2·29-s − 171.·30-s + ⋯ |
| L(s) = 1 | + 0.526·2-s − 1.71·3-s − 0.722·4-s + 1.15·5-s − 0.903·6-s − 1.83·7-s − 0.907·8-s + 1.94·9-s + 0.607·10-s − 0.301·11-s + 1.23·12-s − 0.964·14-s − 1.97·15-s + 0.244·16-s − 1.84·17-s + 1.02·18-s + 0.894·19-s − 0.832·20-s + 3.13·21-s − 0.158·22-s + 1.51·23-s + 1.55·24-s + 0.329·25-s − 1.61·27-s + 1.32·28-s − 0.610·29-s − 1.04·30-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 | \( 1 - 1.49T + 8T^{2} \) |
| 3 | \( 1 + 8.91T + 27T^{2} \) |
| 5 | \( 1 - 12.8T + 125T^{2} \) |
| 7 | \( 1 + 33.8T + 343T^{2} \) |
| 17 | \( 1 + 129.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 95.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 34.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 332.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 229.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 541.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 59.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 681.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 670.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 27.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 173.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 183.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.46e3T + 9.12e5T^{2} \) |
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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
−8.951948491162848420766163015428, −7.14255422326202478752345233624, −6.51642364926435642899124349138, −5.98794539843297468810198561629, −5.36747683098415002786284446138, −4.68590498067493112469783376496, −3.62451236739822093581740198212, −2.51096963511995107099799328128, −0.847131447970898863176780565443, 0,
0.847131447970898863176780565443, 2.51096963511995107099799328128, 3.62451236739822093581740198212, 4.68590498067493112469783376496, 5.36747683098415002786284446138, 5.98794539843297468810198561629, 6.51642364926435642899124349138, 7.14255422326202478752345233624, 8.951948491162848420766163015428
