| L(s) = 1 | − 6·5-s + 20·7-s − 14·11-s + 54·13-s + 66·17-s + 162·19-s − 172·23-s − 89·25-s + 2·29-s − 128·31-s − 120·35-s + 158·37-s − 202·41-s − 298·43-s + 408·47-s + 57·49-s + 690·53-s + 84·55-s + 322·59-s − 298·61-s − 324·65-s + 202·67-s + 700·71-s − 418·73-s − 280·77-s + 744·79-s + 678·83-s + ⋯ |
| L(s) = 1 | − 0.536·5-s + 1.07·7-s − 0.383·11-s + 1.15·13-s + 0.941·17-s + 1.95·19-s − 1.55·23-s − 0.711·25-s + 0.0128·29-s − 0.741·31-s − 0.579·35-s + 0.702·37-s − 0.769·41-s − 1.05·43-s + 1.26·47-s + 0.166·49-s + 1.78·53-s + 0.205·55-s + 0.710·59-s − 0.625·61-s − 0.618·65-s + 0.368·67-s + 1.17·71-s − 0.670·73-s − 0.414·77-s + 1.05·79-s + 0.896·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.351431743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.351431743\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 162 T + p^{3} T^{2} \) |
| 23 | \( 1 + 172 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 + 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 298 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 690 T + p^{3} T^{2} \) |
| 59 | \( 1 - 322 T + p^{3} T^{2} \) |
| 61 | \( 1 + 298 T + p^{3} T^{2} \) |
| 67 | \( 1 - 202 T + p^{3} T^{2} \) |
| 71 | \( 1 - 700 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 - 744 T + p^{3} T^{2} \) |
| 83 | \( 1 - 678 T + p^{3} T^{2} \) |
| 89 | \( 1 - 82 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1122 T + p^{3} T^{2} \) |
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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
−9.461172817110437034272387167114, −8.285983433533332297638445156779, −7.944689292344604026971362519781, −7.17029906189226646608463560567, −5.81070252792695254285890857048, −5.27988848775102058493486471682, −4.07640509352603682369156320485, −3.35227388196277567916199517193, −1.88540147196049520169198670885, −0.832050483919271203609012794384,
0.832050483919271203609012794384, 1.88540147196049520169198670885, 3.35227388196277567916199517193, 4.07640509352603682369156320485, 5.27988848775102058493486471682, 5.81070252792695254285890857048, 7.17029906189226646608463560567, 7.944689292344604026971362519781, 8.285983433533332297638445156779, 9.461172817110437034272387167114
