| L(s) = 1 | + 2·3-s − 7-s + 9-s + 4·11-s + 6·13-s − 2·21-s + 23-s − 5·25-s − 4·27-s + 2·29-s + 2·31-s + 8·33-s + 2·37-s + 12·39-s − 6·41-s + 4·43-s − 2·47-s + 49-s + 14·53-s + 14·59-s + 12·61-s − 63-s − 4·67-s + 2·69-s − 2·73-s − 10·75-s − 4·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.436·21-s + 0.208·23-s − 25-s − 0.769·27-s + 0.371·29-s + 0.359·31-s + 1.39·33-s + 0.328·37-s + 1.92·39-s − 0.937·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s + 1.82·59-s + 1.53·61-s − 0.125·63-s − 0.488·67-s + 0.240·69-s − 0.234·73-s − 1.15·75-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.623016947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.623016947\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−9.515159914427731296818870870826, −8.609158405448298113744621998852, −8.469497890062852666766750103040, −7.25741910042798485514816215730, −6.42281710859097363087217391295, −5.62299425528511656843542759056, −4.00727605955274724755807765716, −3.65861145648623169902938937040, −2.51899572530787692578329494636, −1.27179858482475471748824662576,
1.27179858482475471748824662576, 2.51899572530787692578329494636, 3.65861145648623169902938937040, 4.00727605955274724755807765716, 5.62299425528511656843542759056, 6.42281710859097363087217391295, 7.25741910042798485514816215730, 8.469497890062852666766750103040, 8.609158405448298113744621998852, 9.515159914427731296818870870826
