L(s) = 1 | − 3-s + 4·7-s + 9-s − 2·13-s + 6·17-s − 4·21-s + 4·23-s − 27-s + 2·29-s − 8·31-s + 6·37-s + 2·39-s − 6·41-s + 12·43-s + 12·47-s + 9·49-s − 6·51-s − 10·53-s − 8·59-s + 10·61-s + 4·63-s − 12·67-s − 4·69-s + 8·71-s − 10·73-s + 16·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 0.872·21-s + 0.834·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 1.82·43-s + 1.75·47-s + 9/7·49-s − 0.840·51-s − 1.37·53-s − 1.04·59-s + 1.28·61-s + 0.503·63-s − 1.46·67-s − 0.481·69-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.062473690\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.062473690\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.029457103336535538357682452569, −7.63655571852646626632834186750, −6.98417726522639080944710831631, −5.87116004588496637071643792701, −5.34170379553887299146856715111, −4.72987320531542392704502282013, −3.94165509932302484115605789041, −2.80661991559496432608542478882, −1.72835086757331456987122964680, −0.872533804182944174769366253713,
0.872533804182944174769366253713, 1.72835086757331456987122964680, 2.80661991559496432608542478882, 3.94165509932302484115605789041, 4.72987320531542392704502282013, 5.34170379553887299146856715111, 5.87116004588496637071643792701, 6.98417726522639080944710831631, 7.63655571852646626632834186750, 8.029457103336535538357682452569