| L(s) = 1 | − 2·2-s + 4·4-s + 26·7-s − 8·8-s + 28·11-s + 12·13-s − 52·14-s + 16·16-s + 64·17-s − 60·19-s − 56·22-s + 58·23-s − 24·26-s + 104·28-s − 90·29-s − 128·31-s − 32·32-s − 128·34-s + 236·37-s + 120·38-s − 242·41-s + 362·43-s + 112·44-s − 116·46-s − 226·47-s + 333·49-s + 48·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.40·7-s − 0.353·8-s + 0.767·11-s + 0.256·13-s − 0.992·14-s + 1/4·16-s + 0.913·17-s − 0.724·19-s − 0.542·22-s + 0.525·23-s − 0.181·26-s + 0.701·28-s − 0.576·29-s − 0.741·31-s − 0.176·32-s − 0.645·34-s + 1.04·37-s + 0.512·38-s − 0.921·41-s + 1.28·43-s + 0.383·44-s − 0.371·46-s − 0.701·47-s + 0.970·49-s + 0.128·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.774598730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.774598730\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 64 T + p^{3} T^{2} \) |
| 19 | \( 1 + 60 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 236 T + p^{3} T^{2} \) |
| 41 | \( 1 + 242 T + p^{3} T^{2} \) |
| 43 | \( 1 - 362 T + p^{3} T^{2} \) |
| 47 | \( 1 + 226 T + p^{3} T^{2} \) |
| 53 | \( 1 - 108 T + p^{3} T^{2} \) |
| 59 | \( 1 - 20 T + p^{3} T^{2} \) |
| 61 | \( 1 - 542 T + p^{3} T^{2} \) |
| 67 | \( 1 + 434 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 73 | \( 1 - 632 T + p^{3} T^{2} \) |
| 79 | \( 1 + 720 T + p^{3} T^{2} \) |
| 83 | \( 1 - 478 T + p^{3} T^{2} \) |
| 89 | \( 1 - 490 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1456 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.79229499658099363242185900619, −9.690474398975312718486448574353, −8.795696866813506785574103414311, −8.043811704615687693523727860060, −7.22352316543244588519201618533, −6.06140925045487177154827908423, −4.95397658847046593157868977390, −3.71219632462844697491679711336, −2.06837804278828433221498122687, −1.01262914381224470388917621613,
1.01262914381224470388917621613, 2.06837804278828433221498122687, 3.71219632462844697491679711336, 4.95397658847046593157868977390, 6.06140925045487177154827908423, 7.22352316543244588519201618533, 8.043811704615687693523727860060, 8.795696866813506785574103414311, 9.690474398975312718486448574353, 10.79229499658099363242185900619
