| L(s) = 1 | + 5-s − 3.44·7-s + 2·17-s + 6.89·19-s − 7.44·23-s + 25-s + 1.89·29-s − 1.10·31-s − 3.44·35-s − 6·37-s + 9.89·41-s − 11.7·43-s − 9.44·47-s + 4.89·49-s − 7.79·53-s − 1.10·59-s − 3·61-s + 13.2·67-s + 9.79·71-s + 13.7·73-s + 6.89·79-s − 5.44·83-s + 2·85-s − 2.79·89-s + 6.89·95-s + 2·97-s + 2·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.30·7-s + 0.485·17-s + 1.58·19-s − 1.55·23-s + 0.200·25-s + 0.352·29-s − 0.197·31-s − 0.583·35-s − 0.986·37-s + 1.54·41-s − 1.79·43-s − 1.37·47-s + 0.699·49-s − 1.07·53-s − 0.143·59-s − 0.384·61-s + 1.61·67-s + 1.16·71-s + 1.61·73-s + 0.776·79-s − 0.598·83-s + 0.216·85-s − 0.296·89-s + 0.707·95-s + 0.203·97-s + 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6.89T + 19T^{2} \) |
| 23 | \( 1 + 7.44T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 + 1.10T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 6.89T + 79T^{2} \) |
| 83 | \( 1 + 5.44T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.72625311564088074510738200508, −6.74323440673612004599100064453, −6.37927948760041414273827091542, −5.55024863937292157792583271726, −4.97050102712626707296941390519, −3.72832506110831643065938591819, −3.30580632779638021857339992849, −2.37826302435680562046843035071, −1.29002650284380719422843990297, 0,
1.29002650284380719422843990297, 2.37826302435680562046843035071, 3.30580632779638021857339992849, 3.72832506110831643065938591819, 4.97050102712626707296941390519, 5.55024863937292157792583271726, 6.37927948760041414273827091542, 6.74323440673612004599100064453, 7.72625311564088074510738200508
