| L(s) = 1 | − 2-s + 2.02·3-s + 4-s + 5-s − 2.02·6-s + 1.03·7-s − 8-s + 1.10·9-s − 10-s − 5.03·11-s + 2.02·12-s + 13-s − 1.03·14-s + 2.02·15-s + 16-s − 1.82·17-s − 1.10·18-s − 3.62·19-s + 20-s + 2.08·21-s + 5.03·22-s − 6.12·23-s − 2.02·24-s + 25-s − 26-s − 3.84·27-s + 1.03·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.16·3-s + 0.5·4-s + 0.447·5-s − 0.826·6-s + 0.389·7-s − 0.353·8-s + 0.367·9-s − 0.316·10-s − 1.51·11-s + 0.584·12-s + 0.277·13-s − 0.275·14-s + 0.522·15-s + 0.250·16-s − 0.442·17-s − 0.259·18-s − 0.832·19-s + 0.223·20-s + 0.455·21-s + 1.07·22-s − 1.27·23-s − 0.413·24-s + 0.200·25-s − 0.196·26-s − 0.739·27-s + 0.194·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 2.02T + 3T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + 5.03T + 11T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 + 3.11T + 47T^{2} \) |
| 53 | \( 1 + 7.77T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 7.33T + 61T^{2} \) |
| 67 | \( 1 - 9.11T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 + 3.81T + 79T^{2} \) |
| 83 | \( 1 - 5.06T + 83T^{2} \) |
| 89 | \( 1 + 5.76T + 89T^{2} \) |
| 97 | \( 1 + 0.965T + 97T^{2} \) |
| 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
−8.188792575908704283353184675080, −7.78511865904864782987450462830, −6.78003955546179979376262244694, −5.97358072324547644236143032123, −5.12313857794211520243780516456, −4.11172373715298421462487160633, −3.05218295633883634330497654481, −2.36108165079203516186893442917, −1.72554478047571348235491833441, 0,
1.72554478047571348235491833441, 2.36108165079203516186893442917, 3.05218295633883634330497654481, 4.11172373715298421462487160633, 5.12313857794211520243780516456, 5.97358072324547644236143032123, 6.78003955546179979376262244694, 7.78511865904864782987450462830, 8.188792575908704283353184675080
