L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 2·7-s − 8-s + 9-s − 2·10-s − 11-s − 12-s + 2·14-s − 2·15-s + 16-s − 17-s − 18-s − 2·19-s + 2·20-s + 2·21-s + 22-s − 6·23-s + 24-s − 25-s − 27-s − 2·28-s + 2·29-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s + 0.447·20-s + 0.436·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−9.666890445494638534544971323750, −8.728630371930859666297434411513, −7.80673666939617746581035098942, −6.75686122871305638003743424966, −6.16458234815787488233976486836, −5.44621068688909601380275767472, −4.14668091046034058169448452295, −2.76790035193341092637709593550, −1.67583582527619336581699679256, 0,
1.67583582527619336581699679256, 2.76790035193341092637709593550, 4.14668091046034058169448452295, 5.44621068688909601380275767472, 6.16458234815787488233976486836, 6.75686122871305638003743424966, 7.80673666939617746581035098942, 8.728630371930859666297434411513, 9.666890445494638534544971323750