L(s) = 1 | + 5-s − 7-s − 4·11-s − 6·13-s − 17-s + 4·19-s − 4·23-s + 25-s − 2·29-s + 4·31-s − 35-s − 6·37-s + 2·41-s + 8·47-s + 49-s + 10·53-s − 4·55-s − 8·59-s − 6·61-s − 6·65-s + 6·73-s + 4·77-s − 12·79-s − 12·83-s − 85-s + 6·89-s + 6·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.20·11-s − 1.66·13-s − 0.242·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.169·35-s − 0.986·37-s + 0.312·41-s + 1.16·47-s + 1/7·49-s + 1.37·53-s − 0.539·55-s − 1.04·59-s − 0.768·61-s − 0.744·65-s + 0.702·73-s + 0.455·77-s − 1.35·79-s − 1.31·83-s − 0.108·85-s + 0.635·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7929624439\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7929624439\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.96418925129039, −13.46315417948246, −12.91044720798493, −12.45198032097296, −12.01974514785664, −11.58458663548594, −10.76391660031495, −10.34061311082390, −9.965449828527110, −9.548154612380360, −8.973413752405785, −8.374920214221576, −7.695711611387264, −7.333412903618664, −6.942673984714856, −6.036822570110774, −5.708722416637115, −5.050516430139388, −4.711190450309098, −3.921151375355750, −3.167443199370614, −2.512594516940838, −2.280475479713022, −1.299358013004810, −0.2817446598904505,
0.2817446598904505, 1.299358013004810, 2.280475479713022, 2.512594516940838, 3.167443199370614, 3.921151375355750, 4.711190450309098, 5.050516430139388, 5.708722416637115, 6.036822570110774, 6.942673984714856, 7.333412903618664, 7.695711611387264, 8.374920214221576, 8.973413752405785, 9.548154612380360, 9.965449828527110, 10.34061311082390, 10.76391660031495, 11.58458663548594, 12.01974514785664, 12.45198032097296, 12.91044720798493, 13.46315417948246, 13.96418925129039