| L(s) = 1 | + 1.90·2-s − 2.82·3-s + 1.61·4-s − 3.37·5-s − 5.36·6-s − 1.68·7-s − 0.733·8-s + 4.95·9-s − 6.41·10-s − 2.68·11-s − 4.55·12-s − 1.63·13-s − 3.21·14-s + 9.51·15-s − 4.62·16-s + 0.886·17-s + 9.42·18-s − 0.0521·19-s − 5.44·20-s + 4.76·21-s − 5.11·22-s − 6.14·23-s + 2.06·24-s + 6.37·25-s − 3.10·26-s − 5.51·27-s − 2.72·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 1.62·3-s + 0.807·4-s − 1.50·5-s − 2.18·6-s − 0.638·7-s − 0.259·8-s + 1.65·9-s − 2.02·10-s − 0.810·11-s − 1.31·12-s − 0.453·13-s − 0.857·14-s + 2.45·15-s − 1.15·16-s + 0.214·17-s + 2.22·18-s − 0.0119·19-s − 1.21·20-s + 1.03·21-s − 1.08·22-s − 1.28·23-s + 0.422·24-s + 1.27·25-s − 0.609·26-s − 1.06·27-s − 0.515·28-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
| \(L(1)\) |
\(\approx\) |
\(0.5827990830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5827990830\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.90T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 + 1.68T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 - 0.886T + 17T^{2} \) |
| 19 | \( 1 + 0.0521T + 19T^{2} \) |
| 23 | \( 1 + 6.14T + 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 - 8.21T + 31T^{2} \) |
| 37 | \( 1 + 0.691T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 47 | \( 1 - 9.18T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.69T + 61T^{2} \) |
| 67 | \( 1 + 5.11T + 67T^{2} \) |
| 71 | \( 1 - 8.67T + 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 - 4.37T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 8.65T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−9.455464139295317976338687992212, −8.100548823886669201909990310337, −7.38583093303642566592209186124, −6.51210547380639822072161218560, −5.89164589192150772064251089130, −5.10378503214341940031083837010, −4.40139588334682932684441380947, −3.78965428474357583827348942708, −2.69445618265533993477995064660, −0.43270524972498343399365080946,
0.43270524972498343399365080946, 2.69445618265533993477995064660, 3.78965428474357583827348942708, 4.40139588334682932684441380947, 5.10378503214341940031083837010, 5.89164589192150772064251089130, 6.51210547380639822072161218560, 7.38583093303642566592209186124, 8.100548823886669201909990310337, 9.455464139295317976338687992212
