L(s) = 1 | + 2-s + 3·3-s − 4-s − 2·5-s + 3·6-s + 3·7-s − 3·8-s + 6·9-s − 2·10-s + 3·11-s − 3·12-s + 5·13-s + 3·14-s − 6·15-s − 16-s − 8·17-s + 6·18-s + 4·19-s + 2·20-s + 9·21-s + 3·22-s − 5·23-s − 9·24-s − 25-s + 5·26-s + 9·27-s − 3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.894·5-s + 1.22·6-s + 1.13·7-s − 1.06·8-s + 2·9-s − 0.632·10-s + 0.904·11-s − 0.866·12-s + 1.38·13-s + 0.801·14-s − 1.54·15-s − 1/4·16-s − 1.94·17-s + 1.41·18-s + 0.917·19-s + 0.447·20-s + 1.96·21-s + 0.639·22-s − 1.04·23-s − 1.83·24-s − 1/5·25-s + 0.980·26-s + 1.73·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.918896622\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.918896622\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 503 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−11.22003472564326827475162641185, −9.707744155318615349866592838938, −8.743550895982608798765924402380, −8.477653816930119341239566153030, −7.61354678640816137542686480627, −6.34758581113783796961367002258, −4.68232999229458345747448107651, −4.01103324172883747601833367586, −3.36525387639233754741603332272, −1.77760561249637244183542914407,
1.77760561249637244183542914407, 3.36525387639233754741603332272, 4.01103324172883747601833367586, 4.68232999229458345747448107651, 6.34758581113783796961367002258, 7.61354678640816137542686480627, 8.477653816930119341239566153030, 8.743550895982608798765924402380, 9.707744155318615349866592838938, 11.22003472564326827475162641185