| L(s) = 1 | + 3·3-s − 4·5-s + 7-s + 6·9-s − 2·11-s + 5·13-s − 12·15-s + 4·19-s + 3·21-s − 23-s + 11·25-s + 9·27-s − 3·29-s − 5·31-s − 6·33-s − 4·35-s + 4·37-s + 15·39-s + 5·41-s + 4·43-s − 24·45-s + 11·47-s + 49-s + 8·55-s + 12·57-s + 12·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s + 0.377·7-s + 2·9-s − 0.603·11-s + 1.38·13-s − 3.09·15-s + 0.917·19-s + 0.654·21-s − 0.208·23-s + 11/5·25-s + 1.73·27-s − 0.557·29-s − 0.898·31-s − 1.04·33-s − 0.676·35-s + 0.657·37-s + 2.40·39-s + 0.780·41-s + 0.609·43-s − 3.57·45-s + 1.60·47-s + 1/7·49-s + 1.07·55-s + 1.58·57-s + 1.56·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.760628049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.760628049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.628591547334827806755214944619, −8.216105047172406883892445030005, −7.45646007316228214423029802112, −7.26235357942493417717375956854, −5.74102243942573499536968085340, −4.49780255370785680793150247625, −3.81961122497549643893214757202, −3.36343906798268009384071230258, −2.38500683812798751795154117087, −1.01357002478968873283628561042,
1.01357002478968873283628561042, 2.38500683812798751795154117087, 3.36343906798268009384071230258, 3.81961122497549643893214757202, 4.49780255370785680793150247625, 5.74102243942573499536968085340, 7.26235357942493417717375956854, 7.45646007316228214423029802112, 8.216105047172406883892445030005, 8.628591547334827806755214944619
