| L(s) = 1 | − 1.23·5-s − 7-s − 2.47·11-s − 5.23·13-s + 4.47·17-s + 3.23·19-s + 4·23-s − 3.47·25-s + 4.47·29-s − 6.47·31-s + 1.23·35-s − 4.47·37-s − 0.472·41-s − 2.47·43-s + 1.52·47-s + 49-s − 10·53-s + 3.05·55-s + 4.76·59-s − 6.76·61-s + 6.47·65-s + 4·67-s + 12.9·71-s + 14.9·73-s + 2.47·77-s + 4.94·79-s + 4.76·83-s + ⋯ |
| L(s) = 1 | − 0.552·5-s − 0.377·7-s − 0.745·11-s − 1.45·13-s + 1.08·17-s + 0.742·19-s + 0.834·23-s − 0.694·25-s + 0.830·29-s − 1.16·31-s + 0.208·35-s − 0.735·37-s − 0.0737·41-s − 0.376·43-s + 0.222·47-s + 0.142·49-s − 1.37·53-s + 0.412·55-s + 0.620·59-s − 0.866·61-s + 0.802·65-s + 0.488·67-s + 1.53·71-s + 1.74·73-s + 0.281·77-s + 0.556·79-s + 0.522·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.148060055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.148060055\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 + 6.76T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 4.94T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−8.274456356665318971536017188428, −7.59441071030179057448961562297, −7.24231808989139700875491340724, −6.26906195584521488628022417148, −5.19792256139613805397175975375, −4.94741979568560377917040024331, −3.65448366850550441181475000701, −3.07854810057956205889283035085, −2.07088810042830686614321793020, −0.59114936311307336238315290528,
0.59114936311307336238315290528, 2.07088810042830686614321793020, 3.07854810057956205889283035085, 3.65448366850550441181475000701, 4.94741979568560377917040024331, 5.19792256139613805397175975375, 6.26906195584521488628022417148, 7.24231808989139700875491340724, 7.59441071030179057448961562297, 8.274456356665318971536017188428
