| L(s) = 1 | + 0.596·3-s + 3.74·5-s − 1.10·7-s − 2.64·9-s + 4.05·11-s − 6.28·13-s + 2.23·15-s + 3.42·17-s − 4.55·19-s − 0.661·21-s + 7.12·23-s + 9.00·25-s − 3.36·27-s − 7.58·29-s − 10.1·31-s + 2.41·33-s − 4.14·35-s − 10.9·37-s − 3.74·39-s + 4.41·41-s − 9.01·43-s − 9.89·45-s − 0.789·47-s − 5.77·49-s + 2.04·51-s − 12.5·53-s + 15.1·55-s + ⋯ |
| L(s) = 1 | + 0.344·3-s + 1.67·5-s − 0.418·7-s − 0.881·9-s + 1.22·11-s − 1.74·13-s + 0.576·15-s + 0.831·17-s − 1.04·19-s − 0.144·21-s + 1.48·23-s + 1.80·25-s − 0.648·27-s − 1.40·29-s − 1.82·31-s + 0.421·33-s − 0.701·35-s − 1.79·37-s − 0.600·39-s + 0.688·41-s − 1.37·43-s − 1.47·45-s − 0.115·47-s − 0.824·49-s + 0.286·51-s − 1.72·53-s + 2.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 0.596T + 3T^{2} \) |
| 5 | \( 1 - 3.74T + 5T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 11 | \( 1 - 4.05T + 11T^{2} \) |
| 13 | \( 1 + 6.28T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 7.58T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 + 9.01T + 43T^{2} \) |
| 47 | \( 1 + 0.789T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 8.33T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 + 1.08T + 79T^{2} \) |
| 83 | \( 1 - 8.85T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 7.23T + 97T^{2} \) |
| 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
−7.32564553852480497886828228566, −6.71157718640696023483080821489, −6.16514703499005645488135288472, −5.25900263195764159535695180866, −5.06464537672769951062121254768, −3.64909689566828681400916470336, −3.04824614649757582429510027301, −2.11522219494197680982942559687, −1.61903122415485114802024391472, 0,
1.61903122415485114802024391472, 2.11522219494197680982942559687, 3.04824614649757582429510027301, 3.64909689566828681400916470336, 5.06464537672769951062121254768, 5.25900263195764159535695180866, 6.16514703499005645488135288472, 6.71157718640696023483080821489, 7.32564553852480497886828228566
