| L(s) = 1 | + 5-s − 4.57·7-s − 4.59·11-s − 3.67·13-s − 2.55·17-s + 4.76·19-s − 2.57·23-s + 25-s + 1.91·29-s + 3.46·31-s − 4.57·35-s + 5.46·37-s + 6.64·41-s − 6.55·43-s + 10.0·47-s + 13.9·49-s + 4.03·53-s − 4.59·55-s + 12.2·59-s + 9.59·61-s − 3.67·65-s − 11.3·67-s − 1.67·71-s + 3.12·73-s + 20.9·77-s − 13.2·79-s + 10.7·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.72·7-s − 1.38·11-s − 1.02·13-s − 0.619·17-s + 1.09·19-s − 0.536·23-s + 0.200·25-s + 0.355·29-s + 0.622·31-s − 0.772·35-s + 0.898·37-s + 1.03·41-s − 0.999·43-s + 1.46·47-s + 1.98·49-s + 0.554·53-s − 0.619·55-s + 1.59·59-s + 1.22·61-s − 0.456·65-s − 1.38·67-s − 0.199·71-s + 0.365·73-s + 2.39·77-s − 1.49·79-s + 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057094155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057094155\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4.57T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 + 2.57T + 23T^{2} \) |
| 29 | \( 1 - 1.91T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 9.59T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.809771052461102555259491052768, −7.75109073080838727796063567382, −7.17242523681318787391021845514, −6.37187381553866094297985951841, −5.67560168410219020509415371768, −4.94822058123234328862733457789, −3.87328794669446411044480855668, −2.77472343201376897488406047724, −2.45892540389478995817618660479, −0.57796487025502145222445666744,
0.57796487025502145222445666744, 2.45892540389478995817618660479, 2.77472343201376897488406047724, 3.87328794669446411044480855668, 4.94822058123234328862733457789, 5.67560168410219020509415371768, 6.37187381553866094297985951841, 7.17242523681318787391021845514, 7.75109073080838727796063567382, 8.809771052461102555259491052768
