L(s) = 1 | − 3.16·3-s + 0.310·5-s − 0.631·7-s + 7.03·9-s + 4.43·11-s + 0.596·13-s − 0.984·15-s − 0.105·17-s + 2.07·19-s + 2.00·21-s + 3.00·23-s − 4.90·25-s − 12.7·27-s + 0.274·29-s − 3.01·31-s − 14.0·33-s − 0.196·35-s − 7.47·37-s − 1.88·39-s + 2.70·41-s + 2.91·43-s + 2.18·45-s − 7.70·47-s − 6.60·49-s + 0.333·51-s − 3.12·53-s + 1.37·55-s + ⋯ |
L(s) = 1 | − 1.82·3-s + 0.138·5-s − 0.238·7-s + 2.34·9-s + 1.33·11-s + 0.165·13-s − 0.254·15-s − 0.0255·17-s + 0.475·19-s + 0.436·21-s + 0.626·23-s − 0.980·25-s − 2.46·27-s + 0.0510·29-s − 0.541·31-s − 2.44·33-s − 0.0331·35-s − 1.22·37-s − 0.302·39-s + 0.421·41-s + 0.444·43-s + 0.326·45-s − 1.12·47-s − 0.942·49-s + 0.0467·51-s − 0.429·53-s + 0.185·55-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 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 - 0.310T + 5T^{2} \) |
| 7 | \( 1 + 0.631T + 7T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 - 0.596T + 13T^{2} \) |
| 17 | \( 1 + 0.105T + 17T^{2} \) |
| 19 | \( 1 - 2.07T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 - 0.274T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 1.63T + 61T^{2} \) |
| 67 | \( 1 - 6.02T + 67T^{2} \) |
| 71 | \( 1 - 1.40T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 1.02T + 79T^{2} \) |
| 83 | \( 1 - 3.69T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 + 2.89T + 97T^{2} \) |
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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
−7.19011045131740495607861459112, −6.55327169109318989720334105495, −6.20793252020980062203636092660, −5.43506379508757752330523724995, −4.86459346111021626646586689319, −4.05728754989681325310918912494, −3.36167747346374679250063237629, −1.80901763556065051963975050934, −1.11338766397540841458771282534, 0,
1.11338766397540841458771282534, 1.80901763556065051963975050934, 3.36167747346374679250063237629, 4.05728754989681325310918912494, 4.86459346111021626646586689319, 5.43506379508757752330523724995, 6.20793252020980062203636092660, 6.55327169109318989720334105495, 7.19011045131740495607861459112