| L(s) = 1 | − 2·4-s + 5-s − 3·11-s + 4·16-s − 17-s + 3·19-s − 2·20-s + 25-s + 3·29-s + 10·31-s − 10·37-s + 7·41-s − 43-s + 6·44-s − 6·47-s − 7·49-s + 4·53-s − 3·55-s − 2·59-s + 4·61-s − 8·64-s − 3·67-s + 2·68-s − 4·73-s − 6·76-s + 2·79-s + 4·80-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.904·11-s + 16-s − 0.242·17-s + 0.688·19-s − 0.447·20-s + 1/5·25-s + 0.557·29-s + 1.79·31-s − 1.64·37-s + 1.09·41-s − 0.152·43-s + 0.904·44-s − 0.875·47-s − 49-s + 0.549·53-s − 0.404·55-s − 0.260·59-s + 0.512·61-s − 64-s − 0.366·67-s + 0.242·68-s − 0.468·73-s − 0.688·76-s + 0.225·79-s + 0.447·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683858727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683858727\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.46008666056690, −13.18399332694315, −12.57304242993959, −12.17723385632468, −11.64027258134351, −11.00246294033016, −10.38605935601472, −10.09808315772958, −9.678334631789516, −9.107101964033560, −8.596762883192973, −8.182623862960943, −7.712370954643153, −7.098927944290089, −6.437679986602606, −5.938800180369809, −5.364270191507461, −4.777168373305356, −4.658022838844225, −3.724177054776314, −3.195070337928505, −2.649787377256033, −1.892668333421984, −1.093040577163934, −0.4474504078077285,
0.4474504078077285, 1.093040577163934, 1.892668333421984, 2.649787377256033, 3.195070337928505, 3.724177054776314, 4.658022838844225, 4.777168373305356, 5.364270191507461, 5.938800180369809, 6.437679986602606, 7.098927944290089, 7.712370954643153, 8.182623862960943, 8.596762883192973, 9.107101964033560, 9.678334631789516, 10.09808315772958, 10.38605935601472, 11.00246294033016, 11.64027258134351, 12.17723385632468, 12.57304242993959, 13.18399332694315, 13.46008666056690
