| L(s) = 1 | − 0.571·3-s + 0.953·5-s + 4.33·7-s − 2.67·9-s − 1.47·11-s + 4.03·13-s − 0.545·15-s + 0.0115·17-s − 3.96·19-s − 2.47·21-s − 7.83·23-s − 4.08·25-s + 3.24·27-s − 6.75·29-s − 7.57·31-s + 0.842·33-s + 4.13·35-s − 5.48·37-s − 2.30·39-s + 2.39·41-s − 8.35·43-s − 2.55·45-s − 3.64·47-s + 11.7·49-s − 0.00659·51-s + 11.5·53-s − 1.40·55-s + ⋯ |
| L(s) = 1 | − 0.330·3-s + 0.426·5-s + 1.63·7-s − 0.891·9-s − 0.444·11-s + 1.11·13-s − 0.140·15-s + 0.00279·17-s − 0.909·19-s − 0.540·21-s − 1.63·23-s − 0.817·25-s + 0.624·27-s − 1.25·29-s − 1.36·31-s + 0.146·33-s + 0.699·35-s − 0.902·37-s − 0.369·39-s + 0.373·41-s − 1.27·43-s − 0.380·45-s − 0.532·47-s + 1.68·49-s − 0.000923·51-s + 1.58·53-s − 0.189·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 0.571T + 3T^{2} \) |
| 5 | \( 1 - 0.953T + 5T^{2} \) |
| 7 | \( 1 - 4.33T + 7T^{2} \) |
| 11 | \( 1 + 1.47T + 11T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 - 0.0115T + 17T^{2} \) |
| 19 | \( 1 + 3.96T + 19T^{2} \) |
| 23 | \( 1 + 7.83T + 23T^{2} \) |
| 29 | \( 1 + 6.75T + 29T^{2} \) |
| 31 | \( 1 + 7.57T + 31T^{2} \) |
| 37 | \( 1 + 5.48T + 37T^{2} \) |
| 41 | \( 1 - 2.39T + 41T^{2} \) |
| 43 | \( 1 + 8.35T + 43T^{2} \) |
| 47 | \( 1 + 3.64T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 0.911T + 61T^{2} \) |
| 67 | \( 1 + 3.80T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 + 1.39T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 8.31T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
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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
−8.157063377387908752503202578953, −7.54974219253208429388650292297, −6.40761287834862039426819973137, −5.68567109818663156689302657835, −5.33155982808893021838072636715, −4.31068557815291301613180139009, −3.53605462228378361706929129207, −2.11970510348236414701939609538, −1.66944269580458829375234502453, 0,
1.66944269580458829375234502453, 2.11970510348236414701939609538, 3.53605462228378361706929129207, 4.31068557815291301613180139009, 5.33155982808893021838072636715, 5.68567109818663156689302657835, 6.40761287834862039426819973137, 7.54974219253208429388650292297, 8.157063377387908752503202578953
