L(s) = 1 | − 5-s − 3·11-s + 13-s − 2·17-s + 4·19-s − 4·23-s + 25-s − 4·29-s + 3·37-s − 2·41-s − 2·43-s − 6·47-s + 13·53-s + 3·55-s − 5·59-s + 7·61-s − 65-s − 13·67-s − 5·71-s − 11·73-s + 13·79-s − 4·83-s + 2·85-s + 89-s − 4·95-s − 2·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.904·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.742·29-s + 0.493·37-s − 0.312·41-s − 0.304·43-s − 0.875·47-s + 1.78·53-s + 0.404·55-s − 0.650·59-s + 0.896·61-s − 0.124·65-s − 1.58·67-s − 0.593·71-s − 1.28·73-s + 1.46·79-s − 0.439·83-s + 0.216·85-s + 0.105·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9630284218\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9630284218\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 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
−13.06985863158104, −12.41910926297992, −11.94069431485908, −11.56331933527284, −11.15346011507898, −10.59523925258455, −10.12165015197354, −9.812443214410307, −9.062146300436915, −8.742175517101690, −8.185111925573527, −7.553973661684834, −7.519393933371850, −6.752332825302451, −6.224470545227515, −5.646757682688950, −5.258592849869783, −4.636235927658695, −4.148077307277094, −3.550161599686657, −3.062044662503448, −2.440657525918556, −1.844262909455408, −1.115051773845010, −0.2869033941993053,
0.2869033941993053, 1.115051773845010, 1.844262909455408, 2.440657525918556, 3.062044662503448, 3.550161599686657, 4.148077307277094, 4.636235927658695, 5.258592849869783, 5.646757682688950, 6.224470545227515, 6.752332825302451, 7.519393933371850, 7.553973661684834, 8.185111925573527, 8.742175517101690, 9.062146300436915, 9.812443214410307, 10.12165015197354, 10.59523925258455, 11.15346011507898, 11.56331933527284, 11.94069431485908, 12.41910926297992, 13.06985863158104