| L(s) = 1 | + 3·2-s + 4·4-s + 3·8-s − 5·13-s + 3·16-s − 3·17-s − 6·19-s − 6·23-s + 7·25-s − 15·26-s + 3·29-s + 6·32-s − 9·34-s + 15·37-s − 18·38-s + 9·41-s − 8·43-s − 18·46-s − 7·49-s + 21·50-s − 20·52-s + 6·53-s + 9·58-s − 12·59-s − 61-s + 5·64-s + 6·67-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 1.06·8-s − 1.38·13-s + 3/4·16-s − 0.727·17-s − 1.37·19-s − 1.25·23-s + 7/5·25-s − 2.94·26-s + 0.557·29-s + 1.06·32-s − 1.54·34-s + 2.46·37-s − 2.91·38-s + 1.40·41-s − 1.21·43-s − 2.65·46-s − 49-s + 2.96·50-s − 2.77·52-s + 0.824·53-s + 1.18·58-s − 1.56·59-s − 0.128·61-s + 5/8·64-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.634881853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.634881853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 15 T + 112 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−13.82537752038117811414386352950, −13.09663201674913409202885206212, −12.91113829320065756769769732237, −12.58913598375338648718720352897, −11.87492882753469792665319424840, −11.65594467670936743887824648803, −10.60685065900241612146680633728, −10.56540634359934074886604029254, −9.578556319675191966040292762700, −9.147738395607172997143644102893, −8.121361817311082119247377055501, −7.81619518739241434056211690148, −6.79240429228524573602732606305, −6.38007784589126407921637885043, −5.75306601814164473145468760627, −4.93028819176766711137701447128, −4.49171745133503383136198757094, −4.12193749243753436827426393249, −3.01061525879386457489213948438, −2.30048864547383042155892541730,
2.30048864547383042155892541730, 3.01061525879386457489213948438, 4.12193749243753436827426393249, 4.49171745133503383136198757094, 4.93028819176766711137701447128, 5.75306601814164473145468760627, 6.38007784589126407921637885043, 6.79240429228524573602732606305, 7.81619518739241434056211690148, 8.121361817311082119247377055501, 9.147738395607172997143644102893, 9.578556319675191966040292762700, 10.56540634359934074886604029254, 10.60685065900241612146680633728, 11.65594467670936743887824648803, 11.87492882753469792665319424840, 12.58913598375338648718720352897, 12.91113829320065756769769732237, 13.09663201674913409202885206212, 13.82537752038117811414386352950
