L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 8-s + 9-s + 4·10-s − 2·11-s − 12-s + 4·15-s + 16-s − 8·17-s − 18-s − 19-s − 4·20-s + 2·22-s − 6·23-s + 24-s + 11·25-s − 27-s − 2·29-s − 4·30-s + 8·31-s − 32-s + 2·33-s + 8·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.603·11-s − 0.288·12-s + 1.03·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 0.229·19-s − 0.894·20-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.192·27-s − 0.371·29-s − 0.730·30-s + 1.43·31-s − 0.176·32-s + 0.348·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.56099263587266317490899159222, −6.74795890920795486902997416549, −6.38726310075615636152419705222, −5.09875873161942710020075646652, −4.48121745863743477146109210806, −3.75237564425626419181492888041, −2.79594170357387188561936602870, −1.65146775525669358632030086779, 0, 0,
1.65146775525669358632030086779, 2.79594170357387188561936602870, 3.75237564425626419181492888041, 4.48121745863743477146109210806, 5.09875873161942710020075646652, 6.38726310075615636152419705222, 6.74795890920795486902997416549, 7.56099263587266317490899159222