| L(s) = 1 | − 3-s − 5-s − 7-s − 3·11-s − 8·13-s + 15-s + 4·19-s + 21-s + 8·23-s + 5·25-s + 27-s + 6·29-s − 5·31-s + 3·33-s + 35-s + 8·37-s + 8·39-s + 16·41-s + 12·43-s + 10·47-s − 6·49-s + 9·53-s + 3·55-s − 4·57-s + 5·59-s − 10·61-s + 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 0.904·11-s − 2.21·13-s + 0.258·15-s + 0.917·19-s + 0.218·21-s + 1.66·23-s + 25-s + 0.192·27-s + 1.11·29-s − 0.898·31-s + 0.522·33-s + 0.169·35-s + 1.31·37-s + 1.28·39-s + 2.49·41-s + 1.82·43-s + 1.45·47-s − 6/7·49-s + 1.23·53-s + 0.404·55-s − 0.529·57-s + 0.650·59-s − 1.28·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140133741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140133741\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T - 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−9.781902136945554599033536265168, −9.241997096969628710423291823930, −9.212288569156205225775497691287, −8.803935598067136771225591031170, −7.936635458110483288090868889054, −7.62331958141968081156602846742, −7.44058839681814738828352544225, −7.13780004140830061247801903263, −6.55546476282777777743901221917, −6.07537297320229014400431827855, −5.44830099641157019875692403804, −5.32762618885712249627638982469, −4.70795440166699534158808531588, −4.46708519569930370277086110063, −3.88421380091931557956845279150, −2.89798763996411811623349970824, −2.75192264133369348604027484761, −2.45631239898981412053681327808, −1.09720465747572973934906707699, −0.53715370496901960073564035739,
0.53715370496901960073564035739, 1.09720465747572973934906707699, 2.45631239898981412053681327808, 2.75192264133369348604027484761, 2.89798763996411811623349970824, 3.88421380091931557956845279150, 4.46708519569930370277086110063, 4.70795440166699534158808531588, 5.32762618885712249627638982469, 5.44830099641157019875692403804, 6.07537297320229014400431827855, 6.55546476282777777743901221917, 7.13780004140830061247801903263, 7.44058839681814738828352544225, 7.62331958141968081156602846742, 7.936635458110483288090868889054, 8.803935598067136771225591031170, 9.212288569156205225775497691287, 9.241997096969628710423291823930, 9.781902136945554599033536265168
