| L(s) = 1 | − 0.474·3-s − 4.21·7-s − 2.77·9-s − 3.26·11-s + 0.439·13-s + 2.94·17-s + 1.26·19-s + 2·21-s + 23-s + 2.74·27-s + 5.51·29-s + 0.249·31-s + 1.54·33-s + 8.49·37-s − 0.208·39-s + 2.86·41-s − 1.26·43-s + 7.25·47-s + 10.7·49-s − 1.39·51-s − 9.48·53-s − 0.600·57-s + 0.265·59-s − 1.48·61-s + 11.6·63-s + 3.54·67-s − 0.474·69-s + ⋯ |
| L(s) = 1 | − 0.273·3-s − 1.59·7-s − 0.924·9-s − 0.984·11-s + 0.122·13-s + 0.715·17-s + 0.290·19-s + 0.436·21-s + 0.208·23-s + 0.527·27-s + 1.02·29-s + 0.0447·31-s + 0.269·33-s + 1.39·37-s − 0.0334·39-s + 0.446·41-s − 0.193·43-s + 1.05·47-s + 1.53·49-s − 0.195·51-s − 1.30·53-s − 0.0795·57-s + 0.0345·59-s − 0.189·61-s + 1.47·63-s + 0.433·67-s − 0.0571·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.474T + 3T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 - 0.439T + 13T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 - 0.249T + 31T^{2} \) |
| 37 | \( 1 - 8.49T + 37T^{2} \) |
| 41 | \( 1 - 2.86T + 41T^{2} \) |
| 43 | \( 1 + 1.26T + 43T^{2} \) |
| 47 | \( 1 - 7.25T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 - 0.265T + 59T^{2} \) |
| 61 | \( 1 + 1.48T + 61T^{2} \) |
| 67 | \( 1 - 3.54T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 + 1.08T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.316T + 83T^{2} \) |
| 89 | \( 1 - 8.36T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−7.43888352657053748836514467614, −6.46990314130913911214045451405, −6.08628530254826173327304976246, −5.46056369363408856439227606591, −4.69661764147284089425322140416, −3.65833470383345761363797700400, −2.93558501038108596232900793977, −2.56143788414140763496122970291, −0.943404233443869217495213883420, 0,
0.943404233443869217495213883420, 2.56143788414140763496122970291, 2.93558501038108596232900793977, 3.65833470383345761363797700400, 4.69661764147284089425322140416, 5.46056369363408856439227606591, 6.08628530254826173327304976246, 6.46990314130913911214045451405, 7.43888352657053748836514467614
