| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 2·12-s + 16-s − 6·17-s + 18-s + 2·19-s + 2·24-s − 5·25-s − 4·27-s − 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 2·38-s + 6·41-s + 8·43-s − 12·47-s + 2·48-s − 5·50-s − 12·51-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.408·24-s − 25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 0.288·48-s − 0.707·50-s − 1.68·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16562 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16562 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.93203997356574, −15.37465798001230, −15.15155158131758, −14.35924873858919, −14.06738861581008, −13.50184435655665, −13.01184363650562, −12.59010441071026, −11.66594883268868, −11.27618244937599, −10.72295879176170, −9.863444761110308, −9.215857103171911, −8.945193235668955, −8.040500744654836, −7.637914566721434, −7.016812070547293, −6.231754803897045, −5.639242089597259, −4.879015750862173, −4.042314247301108, −3.681492479977720, −2.825638740604823, −2.241881172581246, −1.581062460810565, 0,
1.581062460810565, 2.241881172581246, 2.825638740604823, 3.681492479977720, 4.042314247301108, 4.879015750862173, 5.639242089597259, 6.231754803897045, 7.016812070547293, 7.637914566721434, 8.040500744654836, 8.945193235668955, 9.215857103171911, 9.863444761110308, 10.72295879176170, 11.27618244937599, 11.66594883268868, 12.59010441071026, 13.01184363650562, 13.50184435655665, 14.06738861581008, 14.35924873858919, 15.15155158131758, 15.37465798001230, 15.93203997356574
