| L(s) = 1 | − 2·3-s + 5-s − 7-s + 9-s + 4·11-s − 2·15-s − 2·17-s + 19-s + 2·21-s + 7·23-s − 4·25-s + 4·27-s + 5·29-s − 9·31-s − 8·33-s − 35-s − 2·37-s − 2·41-s + 43-s + 45-s + 9·47-s + 49-s + 4·51-s − 3·53-s + 4·55-s − 2·57-s − 14·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s − 0.485·17-s + 0.229·19-s + 0.436·21-s + 1.45·23-s − 4/5·25-s + 0.769·27-s + 0.928·29-s − 1.61·31-s − 1.39·33-s − 0.169·35-s − 0.328·37-s − 0.312·41-s + 0.152·43-s + 0.149·45-s + 1.31·47-s + 1/7·49-s + 0.560·51-s − 0.412·53-s + 0.539·55-s − 0.264·57-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.172756110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.172756110\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.96033628709833, −13.53227268430642, −13.02848592536419, −12.40677574599777, −12.02457413264519, −11.61388262487365, −11.04928890667842, −10.63180910029742, −10.19677174228363, −9.452302814396181, −8.931273744311575, −8.856111165366696, −7.768000907567153, −7.126168634181447, −6.799161756624129, −6.150049754213042, −5.837717760577075, −5.293151563637125, −4.599022848848480, −4.162832286629644, −3.304286319914217, −2.804115359701275, −1.783541547696131, −1.256175438998917, −0.4077054728849520,
0.4077054728849520, 1.256175438998917, 1.783541547696131, 2.804115359701275, 3.304286319914217, 4.162832286629644, 4.599022848848480, 5.293151563637125, 5.837717760577075, 6.150049754213042, 6.799161756624129, 7.126168634181447, 7.768000907567153, 8.856111165366696, 8.931273744311575, 9.452302814396181, 10.19677174228363, 10.63180910029742, 11.04928890667842, 11.61388262487365, 12.02457413264519, 12.40677574599777, 13.02848592536419, 13.53227268430642, 13.96033628709833
