| L(s) = 1 | + 1.45·2-s + 0.155·3-s + 0.107·4-s + 0.584·5-s + 0.225·6-s − 7-s − 2.74·8-s − 2.97·9-s + 0.848·10-s − 5.41·11-s + 0.0167·12-s − 3.18·13-s − 1.45·14-s + 0.0907·15-s − 4.20·16-s + 5.06·17-s − 4.32·18-s − 5.11·19-s + 0.0631·20-s − 0.155·21-s − 7.86·22-s + 3.75·23-s − 0.426·24-s − 4.65·25-s − 4.62·26-s − 0.927·27-s − 0.107·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.0895·3-s + 0.0539·4-s + 0.261·5-s + 0.0919·6-s − 0.377·7-s − 0.971·8-s − 0.991·9-s + 0.268·10-s − 1.63·11-s + 0.00483·12-s − 0.882·13-s − 0.388·14-s + 0.0234·15-s − 1.05·16-s + 1.22·17-s − 1.01·18-s − 1.17·19-s + 0.0141·20-s − 0.0338·21-s − 1.67·22-s + 0.783·23-s − 0.0870·24-s − 0.931·25-s − 0.906·26-s − 0.178·27-s − 0.0204·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 889 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 889 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.45T + 2T^{2} \) |
| 3 | \( 1 - 0.155T + 3T^{2} \) |
| 5 | \( 1 - 0.584T + 5T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 - 8.71T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 + 6.65T + 41T^{2} \) |
| 43 | \( 1 + 4.51T + 43T^{2} \) |
| 47 | \( 1 - 9.14T + 47T^{2} \) |
| 53 | \( 1 - 2.10T + 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 3.86T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 6.96T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−9.935129185807366791305079732545, −8.732482818164619983233314098155, −8.119809130374060825670681691044, −6.93800706833010363097386083568, −5.81912522741435831481534443720, −5.34765531535829622638189359548, −4.44225639594743816107132773342, −3.07681642225262103861095867090, −2.58725520200058019940310248264, 0,
2.58725520200058019940310248264, 3.07681642225262103861095867090, 4.44225639594743816107132773342, 5.34765531535829622638189359548, 5.81912522741435831481534443720, 6.93800706833010363097386083568, 8.119809130374060825670681691044, 8.732482818164619983233314098155, 9.935129185807366791305079732545
