L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s − 12-s + 4·13-s + 14-s + 2·15-s + 16-s − 18-s + 6·19-s − 2·20-s + 21-s − 23-s + 24-s − 25-s − 4·26-s − 27-s − 28-s − 6·29-s − 2·30-s − 10·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s + 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s − 0.447·20-s + 0.218·21-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.365·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.423602360911968289207343093832, −8.967192955246074865504328533955, −7.65115015803540650175455814484, −7.42316928282972744645586421519, −6.19152105152837148811171325069, −5.48097069618923915935788224623, −4.05550878791265480207490512911, −3.25219299132934238939732146458, −1.48411961787624910596965721786, 0,
1.48411961787624910596965721786, 3.25219299132934238939732146458, 4.05550878791265480207490512911, 5.48097069618923915935788224623, 6.19152105152837148811171325069, 7.42316928282972744645586421519, 7.65115015803540650175455814484, 8.967192955246074865504328533955, 9.423602360911968289207343093832