L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 11-s − 5·13-s + 14-s + 16-s + 4·17-s + 7·19-s − 22-s + 5·26-s − 28-s − 6·29-s − 32-s − 4·34-s + 4·37-s − 7·38-s − 3·41-s − 11·43-s + 44-s − 47-s + 49-s − 5·52-s − 53-s + 56-s + 6·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.301·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 1.60·19-s − 0.213·22-s + 0.980·26-s − 0.188·28-s − 1.11·29-s − 0.176·32-s − 0.685·34-s + 0.657·37-s − 1.13·38-s − 0.468·41-s − 1.67·43-s + 0.150·44-s − 0.145·47-s + 1/7·49-s − 0.693·52-s − 0.137·53-s + 0.133·56-s + 0.787·58-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.50160592426370673174582283932, −6.90257380824693566677364478275, −6.05974944204480736250084094484, −5.35994783068541243324439365136, −4.72299981886840165833391720298, −3.52205837382236619190595238157, −3.05657132792341723295549228367, −2.05744545230315636611377975368, −1.12957625216972102110433179538, 0,
1.12957625216972102110433179538, 2.05744545230315636611377975368, 3.05657132792341723295549228367, 3.52205837382236619190595238157, 4.72299981886840165833391720298, 5.35994783068541243324439365136, 6.05974944204480736250084094484, 6.90257380824693566677364478275, 7.50160592426370673174582283932