L(s) = 1 | − 2.44·3-s − 3.94·5-s − 0.854·7-s + 2.96·9-s + 0.211·11-s + 0.0490·13-s + 9.63·15-s − 1.50·17-s + 2.95·19-s + 2.08·21-s − 3.34·23-s + 10.5·25-s + 0.0925·27-s + 8.69·29-s − 7.84·31-s − 0.517·33-s + 3.37·35-s − 3.52·37-s − 0.119·39-s − 8.41·41-s + 4.95·43-s − 11.6·45-s − 8.48·47-s − 6.26·49-s + 3.66·51-s + 13.8·53-s − 0.836·55-s + ⋯ |
L(s) = 1 | − 1.40·3-s − 1.76·5-s − 0.322·7-s + 0.987·9-s + 0.0638·11-s + 0.0136·13-s + 2.48·15-s − 0.364·17-s + 0.677·19-s + 0.455·21-s − 0.697·23-s + 2.11·25-s + 0.0178·27-s + 1.61·29-s − 1.40·31-s − 0.0900·33-s + 0.570·35-s − 0.578·37-s − 0.0191·39-s − 1.31·41-s + 0.754·43-s − 1.74·45-s − 1.23·47-s − 0.895·49-s + 0.513·51-s + 1.90·53-s − 0.112·55-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
\(L(1)\) |
\(\approx\) |
\(0.2299889784\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2299889784\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 + 0.854T + 7T^{2} \) |
| 11 | \( 1 - 0.211T + 11T^{2} \) |
| 13 | \( 1 - 0.0490T + 13T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 - 2.95T + 19T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 - 8.69T + 29T^{2} \) |
| 31 | \( 1 + 7.84T + 31T^{2} \) |
| 37 | \( 1 + 3.52T + 37T^{2} \) |
| 41 | \( 1 + 8.41T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 1.88T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 8.33T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 8.56T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−7.71333179385252916177285173513, −6.99521034930409654344595717479, −6.58761969256195098857409778166, −5.71429144533311206049932572679, −4.98022723969785913293685848681, −4.39950014740822631701090523466, −3.66775162012815778092616842006, −2.92300156611236703102559908697, −1.36568958863532111197081402863, −0.27446794495568132097097732937,
0.27446794495568132097097732937, 1.36568958863532111197081402863, 2.92300156611236703102559908697, 3.66775162012815778092616842006, 4.39950014740822631701090523466, 4.98022723969785913293685848681, 5.71429144533311206049932572679, 6.58761969256195098857409778166, 6.99521034930409654344595717479, 7.71333179385252916177285173513