| L(s) = 1 | − 2·5-s + 8·11-s + 6·13-s − 12·17-s − 8·19-s + 2·25-s − 6·29-s + 8·31-s + 2·37-s − 8·43-s − 16·47-s − 2·49-s + 14·53-s − 16·55-s − 6·61-s − 12·65-s + 16·67-s − 24·79-s − 8·83-s + 24·85-s + 16·95-s + 16·97-s + 14·101-s − 16·107-s + 10·109-s − 32·113-s + 32·121-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2.41·11-s + 1.66·13-s − 2.91·17-s − 1.83·19-s + 2/5·25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 1.21·43-s − 2.33·47-s − 2/7·49-s + 1.92·53-s − 2.15·55-s − 0.768·61-s − 1.48·65-s + 1.95·67-s − 2.70·79-s − 0.878·83-s + 2.60·85-s + 1.64·95-s + 1.62·97-s + 1.39·101-s − 1.54·107-s + 0.957·109-s − 3.01·113-s + 2.90·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8720744578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8720744578\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.673696334053679521851080220089, −8.148589144842269016711639958496, −8.125252898525734162345738845524, −7.17167921341114506426061461828, −6.91273493175712294257869481862, −6.71375425939422245946807830270, −6.32919300812062991399649189362, −6.21489734747117650109572719742, −5.81489424590002904063473204570, −4.89162157931904500059247578908, −4.71147686354858537821919516533, −4.17228181895958762948547465419, −4.06616906829066569141482426350, −3.71541901665245164711556887685, −3.38504639685471169939439757536, −2.55160423864811998730976097456, −2.18079949349393398136829352827, −1.51507302033249518543380203513, −1.27527104327676576815016109090, −0.26498777618706456896921579746,
0.26498777618706456896921579746, 1.27527104327676576815016109090, 1.51507302033249518543380203513, 2.18079949349393398136829352827, 2.55160423864811998730976097456, 3.38504639685471169939439757536, 3.71541901665245164711556887685, 4.06616906829066569141482426350, 4.17228181895958762948547465419, 4.71147686354858537821919516533, 4.89162157931904500059247578908, 5.81489424590002904063473204570, 6.21489734747117650109572719742, 6.32919300812062991399649189362, 6.71375425939422245946807830270, 6.91273493175712294257869481862, 7.17167921341114506426061461828, 8.125252898525734162345738845524, 8.148589144842269016711639958496, 8.673696334053679521851080220089
