| L(s) = 1 | − 2-s − 2.37·3-s + 4-s + 3.53·5-s + 2.37·6-s − 2.86·7-s − 8-s + 2.62·9-s − 3.53·10-s − 4.60·11-s − 2.37·12-s + 2.65·13-s + 2.86·14-s − 8.37·15-s + 16-s − 6.58·17-s − 2.62·18-s + 1.84·19-s + 3.53·20-s + 6.79·21-s + 4.60·22-s + 23-s + 2.37·24-s + 7.46·25-s − 2.65·26-s + 0.885·27-s − 2.86·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s + 1.57·5-s + 0.968·6-s − 1.08·7-s − 0.353·8-s + 0.875·9-s − 1.11·10-s − 1.38·11-s − 0.684·12-s + 0.737·13-s + 0.765·14-s − 2.16·15-s + 0.250·16-s − 1.59·17-s − 0.619·18-s + 0.422·19-s + 0.789·20-s + 1.48·21-s + 0.981·22-s + 0.208·23-s + 0.484·24-s + 1.49·25-s − 0.521·26-s + 0.170·27-s − 0.541·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 + 2.86T + 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 + 6.58T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 - 3.71T + 37T^{2} \) |
| 41 | \( 1 - 0.349T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 2.43T + 59T^{2} \) |
| 61 | \( 1 - 8.63T + 61T^{2} \) |
| 67 | \( 1 + 7.42T + 67T^{2} \) |
| 71 | \( 1 + 1.64T + 71T^{2} \) |
| 73 | \( 1 + 4.60T + 73T^{2} \) |
| 79 | \( 1 - 7.32T + 79T^{2} \) |
| 83 | \( 1 + 7.98T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + 4.38T + 97T^{2} \) |
| 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
−7.52145774080277926266341651562, −6.75122579529140199253470530248, −6.33128862294581488588766514757, −5.59173374286635480020711591344, −5.40173307293539489739106919691, −4.19637820530235045071988597550, −2.81120870906001662895955730724, −2.25614432664899981493171716363, −1.02760729263460372720102465612, 0,
1.02760729263460372720102465612, 2.25614432664899981493171716363, 2.81120870906001662895955730724, 4.19637820530235045071988597550, 5.40173307293539489739106919691, 5.59173374286635480020711591344, 6.33128862294581488588766514757, 6.75122579529140199253470530248, 7.52145774080277926266341651562
