| L(s) = 1 | + 16·7-s + 564·11-s + 370·13-s − 1.08e3·17-s − 2.86e3·19-s + 1.58e3·23-s − 1.13e3·29-s − 6.01e3·31-s + 538·37-s − 1.13e4·41-s − 5.44e3·43-s + 1.02e4·47-s − 1.65e4·49-s + 3.47e4·53-s + 2.61e4·59-s + 9.42e3·61-s + 5.11e4·67-s − 1.45e4·71-s + 2.26e4·73-s + 9.02e3·77-s − 9.73e4·79-s − 7.95e3·83-s + 4.79e4·89-s + 5.92e3·91-s − 1.40e5·97-s − 8.53e4·101-s − 1.98e5·103-s + ⋯ |
| L(s) = 1 | + 0.123·7-s + 1.40·11-s + 0.607·13-s − 0.911·17-s − 1.81·19-s + 0.624·23-s − 0.250·29-s − 1.12·31-s + 0.0646·37-s − 1.05·41-s − 0.449·43-s + 0.679·47-s − 0.984·49-s + 1.69·53-s + 0.979·59-s + 0.324·61-s + 1.39·67-s − 0.341·71-s + 0.498·73-s + 0.173·77-s − 1.75·79-s − 0.126·83-s + 0.641·89-s + 0.0749·91-s − 1.51·97-s − 0.832·101-s − 1.84·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 16 T + p^{5} T^{2} \) |
| 11 | \( 1 - 564 T + p^{5} T^{2} \) |
| 13 | \( 1 - 370 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1086 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2860 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1584 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1134 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6016 T + p^{5} T^{2} \) |
| 37 | \( 1 - 538 T + p^{5} T^{2} \) |
| 41 | \( 1 + 11370 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5444 T + p^{5} T^{2} \) |
| 47 | \( 1 - 10296 T + p^{5} T^{2} \) |
| 53 | \( 1 - 34758 T + p^{5} T^{2} \) |
| 59 | \( 1 - 444 p T + p^{5} T^{2} \) |
| 61 | \( 1 - 9422 T + p^{5} T^{2} \) |
| 67 | \( 1 - 51124 T + p^{5} T^{2} \) |
| 71 | \( 1 + 14520 T + p^{5} T^{2} \) |
| 73 | \( 1 - 22678 T + p^{5} T^{2} \) |
| 79 | \( 1 + 97312 T + p^{5} T^{2} \) |
| 83 | \( 1 + 7956 T + p^{5} T^{2} \) |
| 89 | \( 1 - 47910 T + p^{5} T^{2} \) |
| 97 | \( 1 + 140738 T + p^{5} T^{2} \) |
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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
−8.791262718691952227898017333279, −8.421457115008678791862365249059, −6.95728983684912753101784994333, −6.56477418356460704465686056944, −5.49909255131434980080440186861, −4.29606905028607443006236875841, −3.70794056954001487509285479961, −2.26592577486977750707470651992, −1.32454735489900260720184743563, 0,
1.32454735489900260720184743563, 2.26592577486977750707470651992, 3.70794056954001487509285479961, 4.29606905028607443006236875841, 5.49909255131434980080440186861, 6.56477418356460704465686056944, 6.95728983684912753101784994333, 8.421457115008678791862365249059, 8.791262718691952227898017333279
