L(s) = 1 | + 6·11-s − 13-s − 3·17-s − 4·19-s + 3·23-s − 3·29-s + 5·31-s + 10·37-s + 9·41-s − 43-s + 9·53-s − 9·59-s − 11·61-s − 4·67-s − 12·71-s − 10·73-s + 10·79-s + 9·83-s − 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.80·11-s − 0.277·13-s − 0.727·17-s − 0.917·19-s + 0.625·23-s − 0.557·29-s + 0.898·31-s + 1.64·37-s + 1.40·41-s − 0.152·43-s + 1.23·53-s − 1.17·59-s − 1.40·61-s − 0.488·67-s − 1.42·71-s − 1.17·73-s + 1.12·79-s + 0.987·83-s − 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.629718434\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.629718434\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.28984969561959, −12.67880151691307, −12.15696480316957, −11.82684458183704, −11.30382501502464, −10.86537632605946, −10.44893092862244, −9.732645183910576, −9.201807448659789, −9.076919784394219, −8.527716528379515, −7.796227249329106, −7.440104649365744, −6.748591419127302, −6.346030258377582, −6.071262792313385, −5.337094832876440, −4.502422716942608, −4.288842293095450, −3.874185436860548, −2.956158299950891, −2.577613818237959, −1.750227623774565, −1.235618919659125, −0.4887934393616325,
0.4887934393616325, 1.235618919659125, 1.750227623774565, 2.577613818237959, 2.956158299950891, 3.874185436860548, 4.288842293095450, 4.502422716942608, 5.337094832876440, 6.071262792313385, 6.346030258377582, 6.748591419127302, 7.440104649365744, 7.796227249329106, 8.527716528379515, 9.076919784394219, 9.201807448659789, 9.732645183910576, 10.44893092862244, 10.86537632605946, 11.30382501502464, 11.82684458183704, 12.15696480316957, 12.67880151691307, 13.28984969561959