| L(s) = 1 | − 2-s + 0.490·3-s + 4-s + 0.736·5-s − 0.490·6-s − 0.688·7-s − 8-s − 2.75·9-s − 0.736·10-s + 3.29·11-s + 0.490·12-s + 1.45·13-s + 0.688·14-s + 0.361·15-s + 16-s − 2.25·17-s + 2.75·18-s − 3.02·19-s + 0.736·20-s − 0.337·21-s − 3.29·22-s + 3.50·23-s − 0.490·24-s − 4.45·25-s − 1.45·26-s − 2.82·27-s − 0.688·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.283·3-s + 0.5·4-s + 0.329·5-s − 0.200·6-s − 0.260·7-s − 0.353·8-s − 0.919·9-s − 0.233·10-s + 0.994·11-s + 0.141·12-s + 0.403·13-s + 0.184·14-s + 0.0933·15-s + 0.250·16-s − 0.546·17-s + 0.650·18-s − 0.694·19-s + 0.164·20-s − 0.0737·21-s − 0.702·22-s + 0.731·23-s − 0.100·24-s − 0.891·25-s − 0.285·26-s − 0.543·27-s − 0.130·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 0.490T + 3T^{2} \) |
| 5 | \( 1 - 0.736T + 5T^{2} \) |
| 7 | \( 1 + 0.688T + 7T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 23 | \( 1 - 3.50T + 23T^{2} \) |
| 29 | \( 1 + 9.11T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 - 1.88T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.44T + 47T^{2} \) |
| 53 | \( 1 - 5.96T + 53T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 + 1.69T + 61T^{2} \) |
| 67 | \( 1 - 2.98T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.36T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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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
−8.333606180247213019584097735523, −7.46038223303220132543842693562, −6.44619945695085668385974200440, −6.23901760547869656841459373528, −5.21335095093803370196845425045, −4.08622972574084866287227591950, −3.26057123576519887365556888566, −2.34582544788834950462146632830, −1.42708295490530223086844239254, 0,
1.42708295490530223086844239254, 2.34582544788834950462146632830, 3.26057123576519887365556888566, 4.08622972574084866287227591950, 5.21335095093803370196845425045, 6.23901760547869656841459373528, 6.44619945695085668385974200440, 7.46038223303220132543842693562, 8.333606180247213019584097735523
