L(s) = 1 | − 2-s + 4-s − 2·5-s + 4·7-s − 8-s + 2·10-s − 11-s − 13-s − 4·14-s + 16-s + 8·17-s − 6·19-s − 2·20-s + 22-s + 6·23-s − 25-s + 26-s + 4·28-s + 2·29-s − 2·31-s − 32-s − 8·34-s − 8·35-s + 4·37-s + 6·38-s + 2·40-s − 2·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s − 0.353·8-s + 0.632·10-s − 0.301·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.94·17-s − 1.37·19-s − 0.447·20-s + 0.213·22-s + 1.25·23-s − 1/5·25-s + 0.196·26-s + 0.755·28-s + 0.371·29-s − 0.359·31-s − 0.176·32-s − 1.37·34-s − 1.35·35-s + 0.657·37-s + 0.973·38-s + 0.316·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.277635624\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277635624\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.628598354189699697460392586302, −8.068542460543219139102197703686, −7.67936874109208512309952321695, −6.92465036557242612721990389669, −5.73690911092953292829301048396, −4.96209203162457419726183516280, −4.12121782304325588703498643761, −3.05800669196401013024471212590, −1.90420601478738028866725231477, −0.818044508913350002656366627182,
0.818044508913350002656366627182, 1.90420601478738028866725231477, 3.05800669196401013024471212590, 4.12121782304325588703498643761, 4.96209203162457419726183516280, 5.73690911092953292829301048396, 6.92465036557242612721990389669, 7.67936874109208512309952321695, 8.068542460543219139102197703686, 8.628598354189699697460392586302