| L(s) = 1 | − 1.03·3-s + 1.36·5-s + 7-s − 1.93·9-s − 3.61·11-s − 0.433·13-s − 1.41·15-s − 17-s + 7.98·19-s − 1.03·21-s − 1.16·23-s − 3.13·25-s + 5.09·27-s − 5.61·29-s + 8.77·31-s + 3.73·33-s + 1.36·35-s − 2.03·37-s + 0.448·39-s + 2.09·41-s + 4.28·43-s − 2.64·45-s − 3.77·47-s + 49-s + 1.03·51-s − 12.3·53-s − 4.94·55-s + ⋯ |
| L(s) = 1 | − 0.596·3-s + 0.611·5-s + 0.377·7-s − 0.644·9-s − 1.09·11-s − 0.120·13-s − 0.364·15-s − 0.242·17-s + 1.83·19-s − 0.225·21-s − 0.243·23-s − 0.626·25-s + 0.980·27-s − 1.04·29-s + 1.57·31-s + 0.650·33-s + 0.230·35-s − 0.334·37-s + 0.0717·39-s + 0.327·41-s + 0.653·43-s − 0.393·45-s − 0.550·47-s + 0.142·49-s + 0.144·51-s − 1.70·53-s − 0.666·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.03T + 3T^{2} \) |
| 5 | \( 1 - 1.36T + 5T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 + 0.433T + 13T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 + 5.61T + 29T^{2} \) |
| 31 | \( 1 - 8.77T + 31T^{2} \) |
| 37 | \( 1 + 2.03T + 37T^{2} \) |
| 41 | \( 1 - 2.09T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 4.23T + 61T^{2} \) |
| 67 | \( 1 + 0.392T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 5.13T + 73T^{2} \) |
| 79 | \( 1 + 1.81T + 79T^{2} \) |
| 83 | \( 1 + 3.02T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 0.986T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.86561223970494095297814159225, −7.63815491640811837567136162047, −6.42550395396292029347683787737, −5.80954681390420964898534027782, −5.23160202136599311003952877447, −4.60509852463254499377347248010, −3.25428488127814956916713235484, −2.53025453737705424179845195768, −1.38313015777284111533198338154, 0,
1.38313015777284111533198338154, 2.53025453737705424179845195768, 3.25428488127814956916713235484, 4.60509852463254499377347248010, 5.23160202136599311003952877447, 5.80954681390420964898534027782, 6.42550395396292029347683787737, 7.63815491640811837567136162047, 7.86561223970494095297814159225
