| L(s) = 1 | − 2-s − 1.84·3-s + 4-s + 5-s + 1.84·6-s + 3.11·7-s − 8-s + 0.412·9-s − 10-s − 3.92·11-s − 1.84·12-s − 13-s − 3.11·14-s − 1.84·15-s + 16-s + 3.94·17-s − 0.412·18-s + 2.71·19-s + 20-s − 5.76·21-s + 3.92·22-s − 5.82·23-s + 1.84·24-s + 25-s + 26-s + 4.77·27-s + 3.11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.06·3-s + 0.5·4-s + 0.447·5-s + 0.754·6-s + 1.17·7-s − 0.353·8-s + 0.137·9-s − 0.316·10-s − 1.18·11-s − 0.533·12-s − 0.277·13-s − 0.833·14-s − 0.476·15-s + 0.250·16-s + 0.956·17-s − 0.0972·18-s + 0.621·19-s + 0.223·20-s − 1.25·21-s + 0.835·22-s − 1.21·23-s + 0.377·24-s + 0.200·25-s + 0.196·26-s + 0.919·27-s + 0.589·28-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 | \( 1 + 1.84T + 3T^{2} \) |
| 7 | \( 1 - 3.11T + 7T^{2} \) |
| 11 | \( 1 + 3.92T + 11T^{2} \) |
| 17 | \( 1 - 3.94T + 17T^{2} \) |
| 19 | \( 1 - 2.71T + 19T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 - 0.217T + 29T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 - 2.70T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 + 3.67T + 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 + 0.370T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 + 6.37T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 6.62T + 83T^{2} \) |
| 89 | \( 1 + 4.51T + 89T^{2} \) |
| 97 | \( 1 - 3.27T + 97T^{2} \) |
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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
−8.046831311058539452992718204395, −7.50796135821309587616180727739, −6.60944912092138902335155107026, −5.66545023947626462534120601711, −5.35659200935079986412069425884, −4.62461386679492786117420946772, −3.21259994030959992352346199754, −2.17077158876466743215246088559, −1.24411573338059659804220171207, 0,
1.24411573338059659804220171207, 2.17077158876466743215246088559, 3.21259994030959992352346199754, 4.62461386679492786117420946772, 5.35659200935079986412069425884, 5.66545023947626462534120601711, 6.60944912092138902335155107026, 7.50796135821309587616180727739, 8.046831311058539452992718204395
