| L(s) = 1 | − 2·5-s + 2·7-s + 11-s − 2·13-s − 4·17-s + 6·19-s − 25-s + 8·29-s + 8·31-s − 4·35-s + 10·37-s − 8·41-s + 2·43-s − 8·47-s − 3·49-s + 2·53-s − 2·55-s + 12·59-s + 10·61-s + 4·65-s − 12·67-s + 8·71-s + 6·73-s + 2·77-s + 2·79-s + 16·83-s + 8·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 0.301·11-s − 0.554·13-s − 0.970·17-s + 1.37·19-s − 1/5·25-s + 1.48·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s − 1.24·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.269·55-s + 1.56·59-s + 1.28·61-s + 0.496·65-s − 1.46·67-s + 0.949·71-s + 0.702·73-s + 0.227·77-s + 0.225·79-s + 1.75·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533632354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533632354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.451440439716277720463788588234, −8.352307473563578559330404703156, −7.995283121669507299025547417770, −7.08919828768618765963206395693, −6.31116345632917881418704272529, −5.01052020020146862034178505578, −4.52635368073741687498225693994, −3.47375416333532803042906659795, −2.35991906031001647976266238936, −0.890313293886189684730186743974,
0.890313293886189684730186743974, 2.35991906031001647976266238936, 3.47375416333532803042906659795, 4.52635368073741687498225693994, 5.01052020020146862034178505578, 6.31116345632917881418704272529, 7.08919828768618765963206395693, 7.995283121669507299025547417770, 8.352307473563578559330404703156, 9.451440439716277720463788588234
