| L(s) = 1 | + 1.53·3-s + 1.74·5-s − 0.644·9-s + 1.47·11-s − 0.836·13-s + 2.68·15-s + 4.36·17-s − 0.873·19-s + 3.62·23-s − 1.93·25-s − 5.59·27-s + 5.63·29-s + 9.19·31-s + 2.25·33-s + 2.15·37-s − 1.28·39-s − 41-s − 1.50·43-s − 1.12·45-s + 8.53·47-s + 6.70·51-s + 7.12·53-s + 2.57·55-s − 1.34·57-s − 12.9·59-s − 12.0·61-s − 1.46·65-s + ⋯ |
| L(s) = 1 | + 0.886·3-s + 0.782·5-s − 0.214·9-s + 0.443·11-s − 0.231·13-s + 0.693·15-s + 1.05·17-s − 0.200·19-s + 0.754·23-s − 0.387·25-s − 1.07·27-s + 1.04·29-s + 1.65·31-s + 0.392·33-s + 0.354·37-s − 0.205·39-s − 0.156·41-s − 0.228·43-s − 0.168·45-s + 1.24·47-s + 0.938·51-s + 0.978·53-s + 0.347·55-s − 0.177·57-s − 1.68·59-s − 1.54·61-s − 0.181·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.558341460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.558341460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 5 | \( 1 - 1.74T + 5T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 0.836T + 13T^{2} \) |
| 17 | \( 1 - 4.36T + 17T^{2} \) |
| 19 | \( 1 + 0.873T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 9.19T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 43 | \( 1 + 1.50T + 43T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 - 7.12T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 4.82T + 71T^{2} \) |
| 73 | \( 1 + 0.0120T + 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.88840198402777764719333458251, −7.29248490798376662548230810601, −6.30710067903309636554121037659, −5.91305575885123204229300898278, −5.00913136603442480502962746118, −4.25249821794079477178549099420, −3.24226662254494942132070197421, −2.76282350094754118069462466556, −1.90554367713335633353684353092, −0.919613617761374012079029490437,
0.919613617761374012079029490437, 1.90554367713335633353684353092, 2.76282350094754118069462466556, 3.24226662254494942132070197421, 4.25249821794079477178549099420, 5.00913136603442480502962746118, 5.91305575885123204229300898278, 6.30710067903309636554121037659, 7.29248490798376662548230810601, 7.88840198402777764719333458251
