| L(s) = 1 | + 2.82·5-s + 2·7-s + 5.65·11-s + 3.00·25-s − 2.82·29-s + 10·31-s + 5.65·35-s − 3·49-s − 14.1·53-s + 16.0·55-s − 11.3·59-s − 14·73-s + 11.3·77-s + 10·79-s − 5.65·83-s + 2·97-s + 19.7·101-s + 14·103-s − 11.3·107-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + 0.755·7-s + 1.70·11-s + 0.600·25-s − 0.525·29-s + 1.79·31-s + 0.956·35-s − 0.428·49-s − 1.94·53-s + 2.15·55-s − 1.47·59-s − 1.63·73-s + 1.28·77-s + 1.12·79-s − 0.620·83-s + 0.203·97-s + 1.97·101-s + 1.37·103-s − 1.09·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.833585686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.833585686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 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
−9.136398946569171392790795517249, −8.361255339148681707197923814251, −7.44896606746174051922701192764, −6.36523513407901939637797751002, −6.13700566218195458354613235626, −4.99907292809383628661532444156, −4.30943750807639038667710226447, −3.14374695409377479797205561887, −1.92621689418655432325939293744, −1.26177245514393104920732673595,
1.26177245514393104920732673595, 1.92621689418655432325939293744, 3.14374695409377479797205561887, 4.30943750807639038667710226447, 4.99907292809383628661532444156, 6.13700566218195458354613235626, 6.36523513407901939637797751002, 7.44896606746174051922701192764, 8.361255339148681707197923814251, 9.136398946569171392790795517249
