| L(s) = 1 | + 2.75·2-s + 5.59·4-s − 3.97·5-s + 1.71·7-s + 9.90·8-s − 10.9·10-s − 2.05·11-s − 3.29·13-s + 4.73·14-s + 16.1·16-s − 6.66·17-s − 5.33·19-s − 22.2·20-s − 5.67·22-s − 23-s + 10.7·25-s − 9.07·26-s + 9.61·28-s + 29-s − 7.39·31-s + 24.5·32-s − 18.3·34-s − 6.82·35-s − 0.411·37-s − 14.6·38-s − 39.3·40-s + 0.911·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.79·4-s − 1.77·5-s + 0.649·7-s + 3.50·8-s − 3.46·10-s − 0.620·11-s − 0.913·13-s + 1.26·14-s + 4.02·16-s − 1.61·17-s − 1.22·19-s − 4.96·20-s − 1.20·22-s − 0.208·23-s + 2.15·25-s − 1.78·26-s + 1.81·28-s + 0.185·29-s − 1.32·31-s + 4.34·32-s − 3.14·34-s − 1.15·35-s − 0.0676·37-s − 2.38·38-s − 6.21·40-s + 0.142·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 + 6.66T + 17T^{2} \) |
| 19 | \( 1 + 5.33T + 19T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + 0.411T + 37T^{2} \) |
| 41 | \( 1 - 0.911T + 41T^{2} \) |
| 43 | \( 1 + 8.79T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 + 4.39T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 - 6.04T + 67T^{2} \) |
| 71 | \( 1 - 3.14T + 71T^{2} \) |
| 73 | \( 1 + 9.21T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 4.38T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
| 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
−7.52052963685904554160002064431, −6.88890177718643773769762824148, −6.31264815978847637577783417435, −5.01153078825443728434673526034, −4.87624963313677214856917107199, −4.11174799809072421212291030107, −3.58668266631438588528172056731, −2.60576052555633935697761095427, −1.91212612663238645070224445617, 0,
1.91212612663238645070224445617, 2.60576052555633935697761095427, 3.58668266631438588528172056731, 4.11174799809072421212291030107, 4.87624963313677214856917107199, 5.01153078825443728434673526034, 6.31264815978847637577783417435, 6.88890177718643773769762824148, 7.52052963685904554160002064431
