| L(s) = 1 | − 3-s − 2·5-s + 5·7-s − 9-s − 8·11-s + 13-s + 2·15-s + 3·17-s − 2·19-s − 5·21-s + 23-s + 3·25-s − 11·29-s − 2·31-s + 8·33-s − 10·35-s + 6·37-s − 39-s − 4·43-s + 2·45-s − 2·47-s + 9·49-s − 3·51-s + 9·53-s + 16·55-s + 2·57-s + 5·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.88·7-s − 1/3·9-s − 2.41·11-s + 0.277·13-s + 0.516·15-s + 0.727·17-s − 0.458·19-s − 1.09·21-s + 0.208·23-s + 3/5·25-s − 2.04·29-s − 0.359·31-s + 1.39·33-s − 1.69·35-s + 0.986·37-s − 0.160·39-s − 0.609·43-s + 0.298·45-s − 0.291·47-s + 9/7·49-s − 0.420·51-s + 1.23·53-s + 2.15·55-s + 0.264·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 122 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 186 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 17 T + 202 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 60 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.256447982669346368163671144611, −8.127076946012730229522224917624, −7.71339787941334007695996605747, −7.66584508476188592531616769520, −7.19129361024951296503409541927, −6.82788807543498151207696316201, −5.97914551582242472990367206245, −5.60630922824148966745539700575, −5.54608592635643997986852820133, −5.18323632626737688068897390180, −4.55872822282472064507499764032, −4.41966132202833208840966043381, −3.95589789740851047649235296684, −3.17125383857422419706002564499, −2.90826720350117317785188689011, −2.34176164250424344749407496506, −1.66968410110736535011252347448, −1.27561062388972110450253528605, 0, 0,
1.27561062388972110450253528605, 1.66968410110736535011252347448, 2.34176164250424344749407496506, 2.90826720350117317785188689011, 3.17125383857422419706002564499, 3.95589789740851047649235296684, 4.41966132202833208840966043381, 4.55872822282472064507499764032, 5.18323632626737688068897390180, 5.54608592635643997986852820133, 5.60630922824148966745539700575, 5.97914551582242472990367206245, 6.82788807543498151207696316201, 7.19129361024951296503409541927, 7.66584508476188592531616769520, 7.71339787941334007695996605747, 8.127076946012730229522224917624, 8.256447982669346368163671144611
