| L(s) = 1 | + 1.77·2-s − 3-s + 1.14·4-s + 3.50·5-s − 1.77·6-s − 1.50·8-s + 9-s + 6.22·10-s + 11-s − 1.14·12-s + 0.701·13-s − 3.50·15-s − 4.97·16-s + 1.20·17-s + 1.77·18-s + 6.12·19-s + 4.03·20-s + 1.77·22-s + 1.81·23-s + 1.50·24-s + 7.31·25-s + 1.24·26-s − 27-s − 2.68·29-s − 6.22·30-s + 8.64·31-s − 5.81·32-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 0.577·3-s + 0.574·4-s + 1.56·5-s − 0.724·6-s − 0.533·8-s + 0.333·9-s + 1.96·10-s + 0.301·11-s − 0.331·12-s + 0.194·13-s − 0.906·15-s − 1.24·16-s + 0.291·17-s + 0.418·18-s + 1.40·19-s + 0.901·20-s + 0.378·22-s + 0.378·23-s + 0.308·24-s + 1.46·25-s + 0.244·26-s − 0.192·27-s − 0.497·29-s − 1.13·30-s + 1.55·31-s − 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.534199193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.534199193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.77T + 2T^{2} \) |
| 5 | \( 1 - 3.50T + 5T^{2} \) |
| 13 | \( 1 - 0.701T + 13T^{2} \) |
| 17 | \( 1 - 1.20T + 17T^{2} \) |
| 19 | \( 1 - 6.12T + 19T^{2} \) |
| 23 | \( 1 - 1.81T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 - 8.64T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 - 4.55T + 41T^{2} \) |
| 43 | \( 1 + 2.48T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 7.25T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 4.46T + 97T^{2} \) |
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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
−9.464619175434817139783825607163, −8.889307849344660866167532909055, −7.45575388875216793252049494710, −6.49189266304062049395769965825, −5.94380320816409719236803475706, −5.32577961181062684548876327821, −4.67614147665131175302240367217, −3.49403915958785448081885458844, −2.53761638182856829014085305603, −1.25942712656501222998499603107,
1.25942712656501222998499603107, 2.53761638182856829014085305603, 3.49403915958785448081885458844, 4.67614147665131175302240367217, 5.32577961181062684548876327821, 5.94380320816409719236803475706, 6.49189266304062049395769965825, 7.45575388875216793252049494710, 8.889307849344660866167532909055, 9.464619175434817139783825607163
