L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·13-s + 2·14-s + 16-s + 6·17-s − 2·19-s − 5·25-s + 2·26-s − 2·28-s + 6·29-s − 4·31-s − 32-s − 6·34-s + 2·37-s + 2·38-s − 6·41-s + 10·43-s + 12·47-s − 3·49-s + 5·50-s − 2·52-s − 12·53-s + 2·56-s − 6·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 25-s + 0.392·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s + 0.324·38-s − 0.937·41-s + 1.52·43-s + 1.75·47-s − 3/7·49-s + 0.707·50-s − 0.277·52-s − 1.64·53-s + 0.267·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.017689724\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.017689724\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.185719792545246004665453510663, −8.262955010043880759821963787460, −7.61495866909191069709821054220, −6.85457054556699367962615309050, −6.03837711275371548392107302554, −5.28004968616744444473082066349, −4.01426444342608389151630027274, −3.11110186957338669233243757307, −2.12387935125240219535618583660, −0.72040081483209956050869656181,
0.72040081483209956050869656181, 2.12387935125240219535618583660, 3.11110186957338669233243757307, 4.01426444342608389151630027274, 5.28004968616744444473082066349, 6.03837711275371548392107302554, 6.85457054556699367962615309050, 7.61495866909191069709821054220, 8.262955010043880759821963787460, 9.185719792545246004665453510663