| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s − 4·8-s + 4·10-s − 2·11-s − 8·13-s + 5·16-s + 10·17-s − 4·19-s − 6·20-s + 4·22-s − 2·23-s − 5·25-s + 16·26-s + 4·31-s − 6·32-s − 20·34-s + 8·37-s + 8·38-s + 8·40-s + 2·41-s − 6·44-s + 4·46-s + 10·47-s + 10·50-s − 24·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.41·8-s + 1.26·10-s − 0.603·11-s − 2.21·13-s + 5/4·16-s + 2.42·17-s − 0.917·19-s − 1.34·20-s + 0.852·22-s − 0.417·23-s − 25-s + 3.13·26-s + 0.718·31-s − 1.06·32-s − 3.42·34-s + 1.31·37-s + 1.29·38-s + 1.26·40-s + 0.312·41-s − 0.904·44-s + 0.589·46-s + 1.45·47-s + 1.41·50-s − 3.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 3 p T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 101 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 129 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 207 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 26 T + 355 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
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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
−7.62131354390919947494620583538, −7.51398390608683983306335141196, −6.89406788662219584709501861313, −6.86107081388963457720767032075, −6.10188572759833100261551404186, −5.89118354338543739496892010401, −5.54178261197725231872917358725, −5.27742975369708804444415983702, −4.58348046191247645198656524527, −4.49862789472240908485986891463, −3.81468084124013640820181280480, −3.69291415934226054190917048706, −2.86499944919132638862752954716, −2.83295431209764962518859430072, −2.25594491287096537845662703370, −2.07805540155464135333838298770, −1.16623272416659810865035025169, −0.918418781471334153838701053016, 0, 0,
0.918418781471334153838701053016, 1.16623272416659810865035025169, 2.07805540155464135333838298770, 2.25594491287096537845662703370, 2.83295431209764962518859430072, 2.86499944919132638862752954716, 3.69291415934226054190917048706, 3.81468084124013640820181280480, 4.49862789472240908485986891463, 4.58348046191247645198656524527, 5.27742975369708804444415983702, 5.54178261197725231872917358725, 5.89118354338543739496892010401, 6.10188572759833100261551404186, 6.86107081388963457720767032075, 6.89406788662219584709501861313, 7.51398390608683983306335141196, 7.62131354390919947494620583538
