| L(s) = 1 | + 3·3-s + 11·5-s + 7·7-s − 39·11-s − 64·13-s + 33·15-s − 12·17-s + 88·19-s + 21·21-s + 92·23-s + 125·25-s − 27·27-s + 510·29-s + 35·31-s − 117·33-s + 77·35-s + 4·37-s − 192·39-s + 32·41-s − 660·43-s + 298·47-s − 294·49-s − 36·51-s + 717·53-s − 429·55-s + 264·57-s + 217·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.983·5-s + 0.377·7-s − 1.06·11-s − 1.36·13-s + 0.568·15-s − 0.171·17-s + 1.06·19-s + 0.218·21-s + 0.834·23-s + 25-s − 0.192·27-s + 3.26·29-s + 0.202·31-s − 0.617·33-s + 0.371·35-s + 0.0177·37-s − 0.788·39-s + 0.121·41-s − 2.34·43-s + 0.924·47-s − 6/7·49-s − 0.0988·51-s + 1.85·53-s − 1.05·55-s + 0.613·57-s + 0.478·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.244624022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.244624022\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 11 T - 4 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 39 T + 190 T^{2} + 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 12 T - 4769 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 88 T + 885 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 p T - 7 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 255 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 35 T - 28566 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 50637 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 330 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 298 T - 15019 T^{2} - 298 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 717 T + 365212 T^{2} - 717 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 217 T - 158290 T^{2} - 217 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 386 T - 77985 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 906 T + 520073 T^{2} + 906 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 838 T + 313227 T^{2} - 838 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 1325 T + 1262586 T^{2} + 1325 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1163 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 54 T - 702053 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.43904337093615897299193767793, −12.19883523934483721538375945961, −11.75454055746869501856771301609, −10.93221723845187789390038987608, −10.35577752678416938968605696291, −10.14546305783486771270903777149, −9.598559572197554503088466532244, −9.082140688017035713944111026187, −8.304248754114202760190219916500, −8.212228242597398154370728009947, −7.27774006916336697966656434665, −6.93402786080658993338480882145, −6.22777528056974850333732687287, −5.32326491196990260226437268235, −5.01480047606585483937283167813, −4.49503358397661831141401717848, −2.97796877492938693611487125459, −2.87373405596591637629901762538, −1.96956028338962784042241667730, −0.823659039124304091078166114724,
0.823659039124304091078166114724, 1.96956028338962784042241667730, 2.87373405596591637629901762538, 2.97796877492938693611487125459, 4.49503358397661831141401717848, 5.01480047606585483937283167813, 5.32326491196990260226437268235, 6.22777528056974850333732687287, 6.93402786080658993338480882145, 7.27774006916336697966656434665, 8.212228242597398154370728009947, 8.304248754114202760190219916500, 9.082140688017035713944111026187, 9.598559572197554503088466532244, 10.14546305783486771270903777149, 10.35577752678416938968605696291, 10.93221723845187789390038987608, 11.75454055746869501856771301609, 12.19883523934483721538375945961, 12.43904337093615897299193767793
