| L(s) = 1 | − 3-s − 5-s − 5·7-s − 26·9-s − 10·11-s − 13·13-s + 15-s + 93·17-s + 82·19-s + 5·21-s + 192·23-s − 124·25-s + 53·27-s − 106·29-s − 172·31-s + 10·33-s + 5·35-s + 379·37-s + 13·39-s − 148·41-s + 329·43-s + 26·45-s + 631·47-s − 318·49-s − 93·51-s + 160·53-s + 10·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 0.0894·5-s − 0.269·7-s − 0.962·9-s − 0.274·11-s − 0.277·13-s + 0.0172·15-s + 1.32·17-s + 0.990·19-s + 0.0519·21-s + 1.74·23-s − 0.991·25-s + 0.377·27-s − 0.678·29-s − 0.996·31-s + 0.0527·33-s + 0.0241·35-s + 1.68·37-s + 0.0533·39-s − 0.563·41-s + 1.16·43-s + 0.0861·45-s + 1.95·47-s − 0.927·49-s − 0.255·51-s + 0.414·53-s + 0.0245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.479020673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479020673\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 + 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 93 T + p^{3} T^{2} \) |
| 19 | \( 1 - 82 T + p^{3} T^{2} \) |
| 23 | \( 1 - 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 172 T + p^{3} T^{2} \) |
| 37 | \( 1 - 379 T + p^{3} T^{2} \) |
| 41 | \( 1 + 148 T + p^{3} T^{2} \) |
| 43 | \( 1 - 329 T + p^{3} T^{2} \) |
| 47 | \( 1 - 631 T + p^{3} T^{2} \) |
| 53 | \( 1 - 160 T + p^{3} T^{2} \) |
| 59 | \( 1 - 478 T + p^{3} T^{2} \) |
| 61 | \( 1 - 300 T + p^{3} T^{2} \) |
| 67 | \( 1 - 722 T + p^{3} T^{2} \) |
| 71 | \( 1 + 335 T + p^{3} T^{2} \) |
| 73 | \( 1 - 90 T + p^{3} T^{2} \) |
| 79 | \( 1 - 788 T + p^{3} T^{2} \) |
| 83 | \( 1 + 96 T + p^{3} T^{2} \) |
| 89 | \( 1 + 866 T + p^{3} T^{2} \) |
| 97 | \( 1 + 998 T + p^{3} T^{2} \) |
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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
−10.94380230604581157637976489044, −9.802567650855079415159037195789, −9.099782893744442444345235525495, −7.919908741852479378930667334438, −7.18043263731658985938298302545, −5.80194998140012414413386979161, −5.24451636886060418907270089159, −3.66637072604405912300150358104, −2.64129584064889962695520891714, −0.792207331854442004672296517207,
0.792207331854442004672296517207, 2.64129584064889962695520891714, 3.66637072604405912300150358104, 5.24451636886060418907270089159, 5.80194998140012414413386979161, 7.18043263731658985938298302545, 7.919908741852479378930667334438, 9.099782893744442444345235525495, 9.802567650855079415159037195789, 10.94380230604581157637976489044
