L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 5·17-s − 18-s − 2·19-s − 20-s + 21-s − 22-s + 23-s + 24-s − 4·25-s − 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 41 | \( 1 + T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.12062146293561, −15.60820937119894, −15.15619069627405, −14.67726780904120, −13.83445816865554, −13.10373746067717, −12.85647304319360, −12.10758615109728, −11.51396597980169, −11.05266847853735, −10.80165291106484, −9.868349653014357, −9.500398196353226, −8.857662769044121, −8.316855809696195, −7.651444184838129, −6.840763815662750, −6.767137626739585, −5.823567112002288, −5.357416412252870, −4.392329562365335, −3.805575619150327, −3.154853682130229, −1.997220347789467, −1.545854151712534, 0, 0,
1.545854151712534, 1.997220347789467, 3.154853682130229, 3.805575619150327, 4.392329562365335, 5.357416412252870, 5.823567112002288, 6.767137626739585, 6.840763815662750, 7.651444184838129, 8.316855809696195, 8.857662769044121, 9.500398196353226, 9.868349653014357, 10.80165291106484, 11.05266847853735, 11.51396597980169, 12.10758615109728, 12.85647304319360, 13.10373746067717, 13.83445816865554, 14.67726780904120, 15.15619069627405, 15.60820937119894, 16.12062146293561