L(s) = 1 | − 5-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s + 25-s − 10·29-s − 6·37-s − 6·41-s − 4·43-s + 8·47-s + 6·53-s − 4·55-s − 4·59-s − 10·61-s + 2·65-s + 4·67-s − 16·71-s + 14·73-s − 8·79-s − 4·83-s − 2·85-s + 10·89-s − 4·95-s − 10·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.85·29-s − 0.986·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 0.824·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.248·65-s + 0.488·67-s − 1.89·71-s + 1.63·73-s − 0.900·79-s − 0.439·83-s − 0.216·85-s + 1.05·89-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.619567828\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.619567828\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.44481486518317, −12.86762301611687, −12.30373894769434, −11.97710858056510, −11.52511611338151, −11.22230856176402, −10.32452204565856, −10.20121590385182, −9.374256395160005, −9.104941025555434, −8.694029653435991, −7.858589107285616, −7.550419788707328, −7.064516941808607, −6.593447571528268, −5.908961628898296, −5.387036178278727, −4.935321159458848, −4.194241823462716, −3.700715482438555, −3.323367915890390, −2.588786454646560, −1.733472099249038, −1.310756296603244, −0.3873682009279909,
0.3873682009279909, 1.310756296603244, 1.733472099249038, 2.588786454646560, 3.323367915890390, 3.700715482438555, 4.194241823462716, 4.935321159458848, 5.387036178278727, 5.908961628898296, 6.593447571528268, 7.064516941808607, 7.550419788707328, 7.858589107285616, 8.694029653435991, 9.104941025555434, 9.374256395160005, 10.20121590385182, 10.32452204565856, 11.22230856176402, 11.52511611338151, 11.97710858056510, 12.30373894769434, 12.86762301611687, 13.44481486518317