| L(s) = 1 | + 1.78·3-s + (−0.615 − 2.14i)5-s − 2.64·7-s + 0.171·9-s − 0.737i·13-s + (−1.09 − 3.82i)15-s + 5.43i·19-s − 4.71·21-s − 7.48·23-s + (−4.24 + 2.64i)25-s − 5.03·27-s + (1.62 + 5.68i)35-s − 1.31i·39-s + (−0.105 − 0.368i)45-s + 7.00·49-s + ⋯ |
| L(s) = 1 | + 1.02·3-s + (−0.275 − 0.961i)5-s − 0.999·7-s + 0.0571·9-s − 0.204i·13-s + (−0.282 − 0.988i)15-s + 1.24i·19-s − 1.02·21-s − 1.56·23-s + (−0.848 + 0.529i)25-s − 0.969·27-s + (0.275 + 0.961i)35-s − 0.210i·39-s + (−0.0157 − 0.0549i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04908032529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04908032529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.615 + 2.14i)T \) |
| 7 | \( 1 + 2.64T \) |
| good | 3 | \( 1 - 1.78T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 0.737iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.43iT - 19T^{2} \) |
| 23 | \( 1 + 7.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6.45iT - 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 16.9iT - 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.385480475648968372445733759210, −8.448677783641174377979891827039, −8.098406907553087058777646731709, −7.28411601838747298568372035527, −6.09479623590883205832728490498, −5.54798450568862514259733233453, −4.21489323331896254892340425542, −3.67126040486784291091567685588, −2.73132274579538849773632101452, −1.60872514533390451626910844631,
0.01343023243779722351031592582, 2.13162324366914990745193651781, 2.88232864035863213152986496026, 3.52525577234206881318328687096, 4.36277502166619149088575975682, 5.78157087667680551386821134727, 6.49343183093481606241451100977, 7.23571214398914607507976924064, 7.926805812847278022022439098685, 8.740401911291826643235825667902
