| L(s) = 1 | + 3.12·2-s + 4.83·3-s + 1.77·4-s + 5.07·5-s + 15.1·6-s + 2.39·7-s − 19.4·8-s − 3.61·9-s + 15.8·10-s − 30.6·11-s + 8.56·12-s − 2.66·13-s + 7.49·14-s + 24.5·15-s − 75.0·16-s + 13.8·17-s − 11.3·18-s + 97.1·19-s + 8.98·20-s + 11.6·21-s − 95.8·22-s + 26.2·23-s − 94.1·24-s − 99.2·25-s − 8.33·26-s − 148.·27-s + 4.24·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.930·3-s + 0.221·4-s + 0.453·5-s + 1.02·6-s + 0.129·7-s − 0.860·8-s − 0.133·9-s + 0.501·10-s − 0.840·11-s + 0.206·12-s − 0.0568·13-s + 0.143·14-s + 0.422·15-s − 1.17·16-s + 0.197·17-s − 0.148·18-s + 1.17·19-s + 0.100·20-s + 0.120·21-s − 0.929·22-s + 0.237·23-s − 0.800·24-s − 0.794·25-s − 0.0628·26-s − 1.05·27-s + 0.0286·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 3.12T + 8T^{2} \) |
| 3 | \( 1 - 4.83T + 27T^{2} \) |
| 5 | \( 1 - 5.07T + 125T^{2} \) |
| 7 | \( 1 - 2.39T + 343T^{2} \) |
| 11 | \( 1 + 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.66T + 2.19e3T^{2} \) |
| 17 | \( 1 - 13.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 26.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 63.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 23.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 383.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 159.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 63.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 471.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 698.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 356.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 802.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 83.7T + 3.89e5T^{2} \) |
| 79 | \( 1 + 487.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.43e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 519.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−8.426878178102415930091705354492, −7.81367854365021738301666151232, −6.79740880038589953498046434652, −5.72223870818856664276234338424, −5.28297666477246864028802681897, −4.33510710296251012766490495190, −3.23768265075067186753433529956, −2.87241735360440845436500786322, −1.73298974186858536868976540328, 0,
1.73298974186858536868976540328, 2.87241735360440845436500786322, 3.23768265075067186753433529956, 4.33510710296251012766490495190, 5.28297666477246864028802681897, 5.72223870818856664276234338424, 6.79740880038589953498046434652, 7.81367854365021738301666151232, 8.426878178102415930091705354492
