| L(s) = 1 | + 2·2-s + 2·4-s − 9-s − 4·16-s − 8·17-s − 2·18-s − 10·19-s − 25-s − 8·32-s − 16·34-s − 2·36-s − 20·38-s − 2·43-s − 10·49-s − 2·50-s − 8·64-s + 24·67-s − 16·68-s − 20·76-s + 81-s − 12·83-s − 4·86-s + 20·89-s − 20·98-s − 2·100-s − 3·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1/3·9-s − 16-s − 1.94·17-s − 0.471·18-s − 2.29·19-s − 1/5·25-s − 1.41·32-s − 2.74·34-s − 1/3·36-s − 3.24·38-s − 0.304·43-s − 1.42·49-s − 0.282·50-s − 64-s + 2.93·67-s − 1.94·68-s − 2.29·76-s + 1/9·81-s − 1.31·83-s − 0.431·86-s + 2.11·89-s − 2.02·98-s − 1/5·100-s − 0.272·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.833721189099354112677093388174, −8.511837303956780892963774426959, −8.152447020847378149125082250655, −7.28465309598532470280848671908, −6.67225333100919320855192225006, −6.43809935758080971064487134214, −6.08657872101624515828699219466, −5.30545205543166037935734246975, −4.81752207084402549095097320394, −4.33844008032624984171418166297, −3.91942114376096124174149283330, −3.23191447974760057413048040928, −2.31529727169250851598832543664, −2.08231912537489231929557933913, 0,
2.08231912537489231929557933913, 2.31529727169250851598832543664, 3.23191447974760057413048040928, 3.91942114376096124174149283330, 4.33844008032624984171418166297, 4.81752207084402549095097320394, 5.30545205543166037935734246975, 6.08657872101624515828699219466, 6.43809935758080971064487134214, 6.67225333100919320855192225006, 7.28465309598532470280848671908, 8.152447020847378149125082250655, 8.511837303956780892963774426959, 8.833721189099354112677093388174
