| L(s) = 1 | − 3·9-s − 4·11-s + 2·13-s − 2·17-s + 2·19-s − 8·23-s − 5·25-s − 8·31-s + 8·37-s + 6·41-s − 4·43-s + 8·47-s − 7·49-s + 12·53-s + 4·59-s + 2·61-s + 4·67-s − 4·71-s − 6·73-s − 8·79-s + 9·81-s − 12·83-s + 18·89-s − 6·97-s + 12·99-s + 16·101-s + 20·103-s + ⋯ |
| L(s) = 1 | − 9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.458·19-s − 1.66·23-s − 25-s − 1.43·31-s + 1.31·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s − 49-s + 1.64·53-s + 0.520·59-s + 0.256·61-s + 0.488·67-s − 0.474·71-s − 0.702·73-s − 0.900·79-s + 81-s − 1.31·83-s + 1.90·89-s − 0.609·97-s + 1.20·99-s + 1.59·101-s + 1.97·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.071267049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.071267049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−7.73299289868329660071881272678, −7.41507466329920744563879182993, −6.14944202069450756086144920093, −5.85408172415149619019665096787, −5.18976151491910776056356222764, −4.20675355242441383140350699468, −3.52427596168876533000300077387, −2.57924001492494368186860178275, −1.97397974668811029973051561616, −0.48287596893810935025508492253,
0.48287596893810935025508492253, 1.97397974668811029973051561616, 2.57924001492494368186860178275, 3.52427596168876533000300077387, 4.20675355242441383140350699468, 5.18976151491910776056356222764, 5.85408172415149619019665096787, 6.14944202069450756086144920093, 7.41507466329920744563879182993, 7.73299289868329660071881272678
