L(s) = 1 | + 2-s + 4-s + 8-s − 11-s − 2·13-s + 16-s + 6·17-s − 2·19-s − 22-s − 5·25-s − 2·26-s − 6·29-s + 4·31-s + 32-s + 6·34-s + 2·37-s − 2·38-s − 6·41-s − 10·43-s − 44-s − 12·47-s − 5·50-s − 2·52-s − 12·53-s − 6·58-s − 12·59-s + 10·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 0.213·22-s − 25-s − 0.392·26-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 − 0.150·44-s − 1.75·47-s − 0.707·50-s − 0.277·52-s − 1.64·53-s − 0.787·58-s − 1.56·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 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
−7.34269837490323061400629573989, −6.49624194204125351595745838225, −5.96757663993374636611428050958, −5.11622489519355013850379541846, −4.77024972324745778740847357819, −3.65509120254414703818995504468, −3.26123935876568340943979448235, −2.23995632577296827302627462809, −1.46273725464035707570195589728, 0,
1.46273725464035707570195589728, 2.23995632577296827302627462809, 3.26123935876568340943979448235, 3.65509120254414703818995504468, 4.77024972324745778740847357819, 5.11622489519355013850379541846, 5.96757663993374636611428050958, 6.49624194204125351595745838225, 7.34269837490323061400629573989