| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·11-s + 3·13-s + 14-s + 16-s − 17-s + 2·22-s − 3·23-s + 3·26-s + 28-s + 3·29-s − 31-s + 32-s − 34-s + 4·37-s + 10·41-s − 43-s + 2·44-s − 3·46-s − 10·47-s + 49-s + 3·52-s + 53-s + 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.603·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.426·22-s − 0.625·23-s + 0.588·26-s + 0.188·28-s + 0.557·29-s − 0.179·31-s + 0.176·32-s − 0.171·34-s + 0.657·37-s + 1.56·41-s − 0.152·43-s + 0.301·44-s − 0.442·46-s − 1.45·47-s + 1/7·49-s + 0.416·52-s + 0.137·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.029660412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.029660412\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−7.69455673691498121061228247515, −6.79826588031971003456016433080, −6.30939928081133059904825155487, −5.65914241665351043086104495408, −4.88507968809840213360361466569, −4.14298154198731902776549826938, −3.64587306306001343882698456128, −2.67643001611748838418450535308, −1.83234965817256417551014597706, −0.898601354755479293213060635133,
0.898601354755479293213060635133, 1.83234965817256417551014597706, 2.67643001611748838418450535308, 3.64587306306001343882698456128, 4.14298154198731902776549826938, 4.88507968809840213360361466569, 5.65914241665351043086104495408, 6.30939928081133059904825155487, 6.79826588031971003456016433080, 7.69455673691498121061228247515
