| L(s) = 1 | − 1.80·2-s + 1.24·3-s + 1.24·4-s − 2.24·5-s − 2.24·6-s − 1.69·7-s + 1.35·8-s − 1.44·9-s + 4.04·10-s − 1.13·11-s + 1.55·12-s − 6.54·13-s + 3.04·14-s − 2.80·15-s − 4.93·16-s − 3.60·17-s + 2.60·18-s + 4.44·19-s − 2.80·20-s − 2.10·21-s + 2.04·22-s − 3.71·23-s + 1.69·24-s + 0.0489·25-s + 11.7·26-s − 5.54·27-s − 2.10·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.719·3-s + 0.623·4-s − 1.00·5-s − 0.917·6-s − 0.639·7-s + 0.479·8-s − 0.481·9-s + 1.28·10-s − 0.342·11-s + 0.448·12-s − 1.81·13-s + 0.814·14-s − 0.723·15-s − 1.23·16-s − 0.874·17-s + 0.613·18-s + 1.01·19-s − 0.626·20-s − 0.460·21-s + 0.436·22-s − 0.774·23-s + 0.345·24-s + 0.00978·25-s + 2.31·26-s − 1.06·27-s − 0.398·28-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
| \(L(1)\) |
\(\approx\) |
\(0.3839085038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3839085038\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.80T + 2T^{2} \) |
| 3 | \( 1 - 1.24T + 3T^{2} \) |
| 5 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 + 1.13T + 11T^{2} \) |
| 13 | \( 1 + 6.54T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 + 3.71T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 4.39T + 41T^{2} \) |
| 47 | \( 1 + 3.14T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 0.158T + 59T^{2} \) |
| 61 | \( 1 - 0.713T + 61T^{2} \) |
| 67 | \( 1 + 2.09T + 67T^{2} \) |
| 71 | \( 1 - 3.72T + 71T^{2} \) |
| 73 | \( 1 + 1.43T + 73T^{2} \) |
| 79 | \( 1 + 0.0609T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 3.50T + 97T^{2} \) |
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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
−9.142186312582807584904509388238, −8.469196052907144546716462450890, −7.81698743367539954901972329680, −7.37692648358527437884318697690, −6.46206055683122965932303134810, −5.03729536728359239158677515489, −4.19915599037581549161596894791, −3.00123712922808238790919492155, −2.28566201773003561363661374490, −0.46127665438387725214585069244,
0.46127665438387725214585069244, 2.28566201773003561363661374490, 3.00123712922808238790919492155, 4.19915599037581549161596894791, 5.03729536728359239158677515489, 6.46206055683122965932303134810, 7.37692648358527437884318697690, 7.81698743367539954901972329680, 8.469196052907144546716462450890, 9.142186312582807584904509388238
