| L(s) = 1 | − 0.599·2-s − 1.15·3-s − 1.64·4-s + 5-s + 0.691·6-s − 3.77·7-s + 2.18·8-s − 1.66·9-s − 0.599·10-s + 3.97·11-s + 1.89·12-s − 4.01·13-s + 2.26·14-s − 1.15·15-s + 1.97·16-s + 17-s + 0.999·18-s + 2.07·19-s − 1.64·20-s + 4.35·21-s − 2.38·22-s − 2.14·23-s − 2.51·24-s + 25-s + 2.40·26-s + 5.38·27-s + 6.18·28-s + ⋯ |
| L(s) = 1 | − 0.423·2-s − 0.666·3-s − 0.820·4-s + 0.447·5-s + 0.282·6-s − 1.42·7-s + 0.771·8-s − 0.555·9-s − 0.189·10-s + 1.19·11-s + 0.546·12-s − 1.11·13-s + 0.604·14-s − 0.297·15-s + 0.493·16-s + 0.242·17-s + 0.235·18-s + 0.474·19-s − 0.366·20-s + 0.949·21-s − 0.508·22-s − 0.446·23-s − 0.514·24-s + 0.200·25-s + 0.472·26-s + 1.03·27-s + 1.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 0.599T + 2T^{2} \) |
| 3 | \( 1 + 1.15T + 3T^{2} \) |
| 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 - 3.97T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 19 | \( 1 - 2.07T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 + 8.52T + 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 - 1.34T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 9.98T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 73 | \( 1 - 8.99T + 73T^{2} \) |
| 79 | \( 1 + 9.27T + 79T^{2} \) |
| 83 | \( 1 - 9.10T + 83T^{2} \) |
| 89 | \( 1 - 4.00T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−7.56747517619135675714720942563, −7.07107896951346796601690260898, −6.15767983859035461840389309363, −5.69875269740136654435795456496, −4.98469086791305477063853233342, −3.99442294489755299807741333862, −3.35362881048186523264036388780, −2.24192990903809040295157411733, −0.912021356295450662117249938474, 0,
0.912021356295450662117249938474, 2.24192990903809040295157411733, 3.35362881048186523264036388780, 3.99442294489755299807741333862, 4.98469086791305477063853233342, 5.69875269740136654435795456496, 6.15767983859035461840389309363, 7.07107896951346796601690260898, 7.56747517619135675714720942563
