| L(s) = 1 | − 3·5-s − 7-s + 4·11-s + 3·13-s − 23-s + 4·25-s − 29-s + 2·31-s + 3·35-s − 5·37-s − 5·41-s + 7·43-s − 3·47-s + 49-s − 12·53-s − 12·55-s − 2·59-s − 6·61-s − 9·65-s + 12·67-s + 10·71-s − 4·77-s − 4·79-s + 4·83-s − 10·89-s − 3·91-s + 19·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.20·11-s + 0.832·13-s − 0.208·23-s + 4/5·25-s − 0.185·29-s + 0.359·31-s + 0.507·35-s − 0.821·37-s − 0.780·41-s + 1.06·43-s − 0.437·47-s + 1/7·49-s − 1.64·53-s − 1.61·55-s − 0.260·59-s − 0.768·61-s − 1.11·65-s + 1.46·67-s + 1.18·71-s − 0.455·77-s − 0.450·79-s + 0.439·83-s − 1.05·89-s − 0.314·91-s + 1.92·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−15.65914576650016, −15.44214807728753, −14.64634510842352, −14.17158852242057, −13.67254340962598, −12.92587610269598, −12.32646710868328, −12.02381457861717, −11.24411475151981, −11.14546654205384, −10.29986159929324, −9.597990531370029, −9.039320013087410, −8.494402000662264, −7.939974318040495, −7.390174134495378, −6.567178130612782, −6.388514693468719, −5.433551361632334, −4.637224855362053, −3.947630911030884, −3.627081274760552, −2.963982884483002, −1.803657749006983, −0.9778450799842870, 0,
0.9778450799842870, 1.803657749006983, 2.963982884483002, 3.627081274760552, 3.947630911030884, 4.637224855362053, 5.433551361632334, 6.388514693468719, 6.567178130612782, 7.390174134495378, 7.939974318040495, 8.494402000662264, 9.039320013087410, 9.597990531370029, 10.29986159929324, 11.14546654205384, 11.24411475151981, 12.02381457861717, 12.32646710868328, 12.92587610269598, 13.67254340962598, 14.17158852242057, 14.64634510842352, 15.44214807728753, 15.65914576650016
