| L(s) = 1 | − 3-s − 2·5-s − 5·7-s + 3·9-s − 11-s + 6·13-s + 2·15-s + 2·17-s + 4·19-s + 5·21-s − 4·23-s + 5·25-s − 8·27-s − 14·29-s − 8·31-s + 33-s + 10·35-s − 12·37-s − 6·39-s − 16·41-s − 16·43-s − 6·45-s − 10·47-s + 18·49-s − 2·51-s − 14·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.88·7-s + 9-s − 0.301·11-s + 1.66·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s + 1.09·21-s − 0.834·23-s + 25-s − 1.53·27-s − 2.59·29-s − 1.43·31-s + 0.174·33-s + 1.69·35-s − 1.97·37-s − 0.960·39-s − 2.49·41-s − 2.43·43-s − 0.894·45-s − 1.45·47-s + 18/7·49-s − 0.280·51-s − 1.92·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−9.475239784673377497441863918672, −9.329933630163556129072047486157, −8.630049228452528259932130671459, −8.390736581366374744341806203498, −7.67582800010611480841370463373, −7.49465592859873800125967126719, −6.92176266594279234629306708142, −6.67883005791140474158658675120, −6.23717570792415287986264649643, −5.77518303391107513240947367317, −5.22223467949035846085669114292, −5.04124582972576391114636920272, −4.06858744375042879372624427463, −3.52928926040052736987646460800, −3.37465427735161400154629537640, −3.37374701955847795608371540461, −1.68425286801187229954033408362, −1.66272708077020189325072904002, 0, 0,
1.66272708077020189325072904002, 1.68425286801187229954033408362, 3.37374701955847795608371540461, 3.37465427735161400154629537640, 3.52928926040052736987646460800, 4.06858744375042879372624427463, 5.04124582972576391114636920272, 5.22223467949035846085669114292, 5.77518303391107513240947367317, 6.23717570792415287986264649643, 6.67883005791140474158658675120, 6.92176266594279234629306708142, 7.49465592859873800125967126719, 7.67582800010611480841370463373, 8.390736581366374744341806203498, 8.630049228452528259932130671459, 9.329933630163556129072047486157, 9.475239784673377497441863918672
