| L(s) = 1 | − 2.14i·3-s − 2.23·5-s + 9.06i·7-s + 4.41·9-s − 4.28i·11-s + 9.41·13-s + 4.78i·15-s + 18·17-s + 36.2i·19-s + 19.4·21-s − 22.9i·23-s + 5.00·25-s − 28.7i·27-s + 44.8·29-s − 35.2i·31-s + ⋯ |
| L(s) = 1 | − 0.713i·3-s − 0.447·5-s + 1.29i·7-s + 0.490·9-s − 0.389i·11-s + 0.724·13-s + 0.319i·15-s + 1.05·17-s + 1.90i·19-s + 0.924·21-s − 0.996i·23-s + 0.200·25-s − 1.06i·27-s + 1.54·29-s − 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.66521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66521\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 3 | \( 1 + 2.14iT - 9T^{2} \) |
| 7 | \( 1 - 9.06iT - 49T^{2} \) |
| 11 | \( 1 + 4.28iT - 121T^{2} \) |
| 13 | \( 1 - 9.41T + 169T^{2} \) |
| 17 | \( 1 - 18T + 289T^{2} \) |
| 19 | \( 1 - 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 22.9iT - 529T^{2} \) |
| 29 | \( 1 - 44.8T + 841T^{2} \) |
| 31 | \( 1 + 35.2iT - 961T^{2} \) |
| 37 | \( 1 + 6.58T + 1.36e3T^{2} \) |
| 41 | \( 1 - 52.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 28.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 90.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 52.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 17.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 33.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 20.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 91.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 42.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 22.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 47.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + 160.T + 9.40e3T^{2} \) |
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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
−11.71685110242750923372804259420, −10.52900933636185738208792067480, −9.496265875652880865026323968552, −8.256658108359328561975670986852, −7.87207188036710691323161784764, −6.38330528048468494767023558542, −5.76122042257327485584309035972, −4.19658452157222377852499453232, −2.78595069915151970540988332848, −1.27398856153856427168139671304,
1.02442044487613205647319190127, 3.31139074840849332981094376640, 4.22599017357692908189155399042, 5.10480180037302289148853039960, 6.82666060148061990724109302561, 7.42957995482404087322484739708, 8.668563976369025947209238925766, 9.748525914257226471200411776665, 10.48154691811003791076172863382, 11.16156865583161486850242179458
