| L(s) = 1 | − 0.959·2-s − 1.07·4-s + 4.21·5-s − 1.12·7-s + 2.95·8-s − 4.04·10-s − 3.26·11-s + 5.17·13-s + 1.07·14-s − 0.676·16-s − 3.37·17-s + 2.36·19-s − 4.55·20-s + 3.13·22-s + 6.68·23-s + 12.7·25-s − 4.96·26-s + 1.20·28-s + 1.71·29-s − 5.26·32-s + 3.23·34-s − 4.72·35-s − 4.47·37-s − 2.26·38-s + 12.4·40-s − 0.225·41-s + 4.92·43-s + ⋯ |
| L(s) = 1 | − 0.678·2-s − 0.539·4-s + 1.88·5-s − 0.423·7-s + 1.04·8-s − 1.27·10-s − 0.984·11-s + 1.43·13-s + 0.287·14-s − 0.169·16-s − 0.818·17-s + 0.542·19-s − 1.01·20-s + 0.667·22-s + 1.39·23-s + 2.55·25-s − 0.972·26-s + 0.228·28-s + 0.319·29-s − 0.929·32-s + 0.555·34-s − 0.799·35-s − 0.735·37-s − 0.368·38-s + 1.97·40-s − 0.0352·41-s + 0.751·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.846400978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.846400978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 0.959T + 2T^{2} \) |
| 5 | \( 1 - 4.21T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 - 5.17T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 - 2.36T + 19T^{2} \) |
| 23 | \( 1 - 6.68T + 23T^{2} \) |
| 29 | \( 1 - 1.71T + 29T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 0.225T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 4.54T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 2.57T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 + 2.10T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.933014478445952198470648298865, −7.00731471402217492070252736551, −6.44569509729961190929005539629, −5.61012447061712328213948185443, −5.24267210549382753881103106662, −4.38975036683958483830902638754, −3.27498748522638203917748051586, −2.48497876545747065873824132729, −1.56535779090481607451690794192, −0.792107746009594666050215558341,
0.792107746009594666050215558341, 1.56535779090481607451690794192, 2.48497876545747065873824132729, 3.27498748522638203917748051586, 4.38975036683958483830902638754, 5.24267210549382753881103106662, 5.61012447061712328213948185443, 6.44569509729961190929005539629, 7.00731471402217492070252736551, 7.933014478445952198470648298865
