| L(s) = 1 | − 25·5-s − 80·7-s − 684·11-s − 978·13-s + 862·17-s + 916·19-s + 1.55e3·23-s + 625·25-s + 7.31e3·29-s − 9.31e3·31-s + 2.00e3·35-s − 8.82e3·37-s + 3.28e3·41-s + 7.55e3·43-s + 5.96e3·47-s − 1.04e4·49-s + 8.69e3·53-s + 1.71e4·55-s + 4.20e4·59-s + 3.75e4·61-s + 2.44e4·65-s + 2.93e4·67-s − 8.44e4·71-s − 4.65e4·73-s + 5.47e4·77-s + 2.67e4·79-s + 7.95e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.617·7-s − 1.70·11-s − 1.60·13-s + 0.723·17-s + 0.582·19-s + 0.611·23-s + 1/5·25-s + 1.61·29-s − 1.74·31-s + 0.275·35-s − 1.05·37-s + 0.305·41-s + 0.623·43-s + 0.393·47-s − 0.619·49-s + 0.425·53-s + 0.762·55-s + 1.57·59-s + 1.29·61-s + 0.717·65-s + 0.798·67-s − 1.98·71-s − 1.02·73-s + 1.05·77-s + 0.482·79-s + 0.126·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9767463159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9767463159\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 80 T + p^{5} T^{2} \) |
| 11 | \( 1 + 684 T + p^{5} T^{2} \) |
| 13 | \( 1 + 978 T + p^{5} T^{2} \) |
| 17 | \( 1 - 862 T + p^{5} T^{2} \) |
| 19 | \( 1 - 916 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1552 T + p^{5} T^{2} \) |
| 29 | \( 1 - 7314 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9312 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8826 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3286 T + p^{5} T^{2} \) |
| 43 | \( 1 - 7556 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5960 T + p^{5} T^{2} \) |
| 53 | \( 1 - 8698 T + p^{5} T^{2} \) |
| 59 | \( 1 - 42036 T + p^{5} T^{2} \) |
| 61 | \( 1 - 37518 T + p^{5} T^{2} \) |
| 67 | \( 1 - 29324 T + p^{5} T^{2} \) |
| 71 | \( 1 + 84408 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46550 T + p^{5} T^{2} \) |
| 79 | \( 1 - 26752 T + p^{5} T^{2} \) |
| 83 | \( 1 - 7956 T + p^{5} T^{2} \) |
| 89 | \( 1 + 59674 T + p^{5} T^{2} \) |
| 97 | \( 1 - 136898 T + p^{5} T^{2} \) |
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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
−10.39409989837979957692029650450, −9.927152089219527525531912136372, −8.731629321587968796663495060005, −7.60122988882278432912488035545, −7.11485135672296044518683425160, −5.55374430410425713691613614035, −4.84094757153196832589415472453, −3.31624577716782782197589273971, −2.43744976966071119026030521207, −0.50649047780436576600785217884,
0.50649047780436576600785217884, 2.43744976966071119026030521207, 3.31624577716782782197589273971, 4.84094757153196832589415472453, 5.55374430410425713691613614035, 7.11485135672296044518683425160, 7.60122988882278432912488035545, 8.731629321587968796663495060005, 9.927152089219527525531912136372, 10.39409989837979957692029650450
