| L(s) = 1 | + 1.86·2-s − 2.42·3-s + 1.46·4-s + 1.27·5-s − 4.51·6-s + 1.26·7-s − 0.996·8-s + 2.87·9-s + 2.37·10-s − 6.14·11-s − 3.55·12-s + 6.48·13-s + 2.36·14-s − 3.09·15-s − 4.78·16-s − 2.03·17-s + 5.35·18-s − 3.13·19-s + 1.86·20-s − 3.07·21-s − 11.4·22-s − 0.564·23-s + 2.41·24-s − 3.37·25-s + 12.0·26-s + 0.291·27-s + 1.85·28-s + ⋯ |
| L(s) = 1 | + 1.31·2-s − 1.39·3-s + 0.732·4-s + 0.569·5-s − 1.84·6-s + 0.479·7-s − 0.352·8-s + 0.959·9-s + 0.750·10-s − 1.85·11-s − 1.02·12-s + 1.79·13-s + 0.631·14-s − 0.797·15-s − 1.19·16-s − 0.494·17-s + 1.26·18-s − 0.718·19-s + 0.417·20-s − 0.671·21-s − 2.43·22-s − 0.117·23-s + 0.493·24-s − 0.675·25-s + 2.36·26-s + 0.0561·27-s + 0.351·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.86T + 2T^{2} \) |
| 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 6.14T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 + 0.564T + 23T^{2} \) |
| 29 | \( 1 - 0.575T + 29T^{2} \) |
| 31 | \( 1 - 0.511T + 31T^{2} \) |
| 37 | \( 1 + 4.97T + 37T^{2} \) |
| 41 | \( 1 - 0.529T + 41T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.15T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 61 | \( 1 - 3.75T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 2.23T + 71T^{2} \) |
| 73 | \( 1 - 0.806T + 73T^{2} \) |
| 79 | \( 1 + 7.98T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−8.735533928386574557481662619415, −8.001472614566123174082709669425, −6.66217508456872322150706837023, −6.13578344664915695819549765785, −5.50313825587169912119578537338, −4.99334465004997840150122873521, −4.17991795422463720143171607583, −3.02586447316154451417275322018, −1.79266537514806681497215846141, 0,
1.79266537514806681497215846141, 3.02586447316154451417275322018, 4.17991795422463720143171607583, 4.99334465004997840150122873521, 5.50313825587169912119578537338, 6.13578344664915695819549765785, 6.66217508456872322150706837023, 8.001472614566123174082709669425, 8.735533928386574557481662619415
