| L(s) = 1 | − 3·5-s + 7-s + 3.79·11-s − 13-s − 3.79·17-s + 19-s − 4.58·23-s + 4·25-s − 3.79·29-s + 7.37·31-s − 3·35-s + 5·37-s + 3.79·41-s + 2·43-s − 10.5·47-s + 49-s + 8.37·53-s − 11.3·55-s − 12.1·59-s − 61-s + 3·65-s − 9.37·67-s + 12.1·71-s + 16.3·73-s + 3.79·77-s − 10·79-s − 14.3·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 1.14·11-s − 0.277·13-s − 0.919·17-s + 0.229·19-s − 0.955·23-s + 0.800·25-s − 0.704·29-s + 1.32·31-s − 0.507·35-s + 0.821·37-s + 0.592·41-s + 0.304·43-s − 1.54·47-s + 0.142·49-s + 1.15·53-s − 1.53·55-s − 1.58·59-s − 0.128·61-s + 0.372·65-s − 1.14·67-s + 1.44·71-s + 1.91·73-s + 0.432·77-s − 1.12·79-s − 1.57·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 3T + 5T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 23 | \( 1 + 4.58T + 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 - 8.37T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 9.37T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 7.58T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.019719056045307954021325888623, −7.25314307276747735944767511874, −6.59998835000921810157139539036, −5.81301493208881776699562282671, −4.62729378536117130988037843585, −4.23876214820757577371486467760, −3.52317966597678152123509557470, −2.44794237707911530887908187109, −1.26796192541073129789232848734, 0,
1.26796192541073129789232848734, 2.44794237707911530887908187109, 3.52317966597678152123509557470, 4.23876214820757577371486467760, 4.62729378536117130988037843585, 5.81301493208881776699562282671, 6.59998835000921810157139539036, 7.25314307276747735944767511874, 8.019719056045307954021325888623
