| L(s) = 1 | + 2·3-s − 3·4-s − 2·7-s + 3·9-s + 8·11-s − 6·12-s + 5·16-s − 4·21-s − 6·25-s + 4·27-s + 6·28-s + 16·33-s − 9·36-s + 6·37-s + 4·41-s − 24·44-s + 10·48-s + 3·49-s + 12·53-s − 6·63-s − 3·64-s + 8·67-s − 12·73-s − 12·75-s − 16·77-s + 5·81-s − 24·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s − 0.755·7-s + 9-s + 2.41·11-s − 1.73·12-s + 5/4·16-s − 0.872·21-s − 6/5·25-s + 0.769·27-s + 1.13·28-s + 2.78·33-s − 3/2·36-s + 0.986·37-s + 0.624·41-s − 3.61·44-s + 1.44·48-s + 3/7·49-s + 1.64·53-s − 0.755·63-s − 3/8·64-s + 0.977·67-s − 1.40·73-s − 1.38·75-s − 1.82·77-s + 5/9·81-s − 2.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141186164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141186164\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 37 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.672365196683144779961098572501, −8.208681846377239693918583479320, −7.44557030498848387492737405058, −7.25047783802838427330028450541, −6.63361830396013314539378177155, −6.05511775830054759433995111157, −5.81425642275690049568553481297, −4.95955492048954303831149329828, −4.24941633754205518250257909295, −4.13559084050773741974089362056, −3.71318515089069514274254472398, −3.21417170816838145432618937923, −2.41395863427134249641033263863, −1.56301279824709801760344144009, −0.76577195414570462258554927346,
0.76577195414570462258554927346, 1.56301279824709801760344144009, 2.41395863427134249641033263863, 3.21417170816838145432618937923, 3.71318515089069514274254472398, 4.13559084050773741974089362056, 4.24941633754205518250257909295, 4.95955492048954303831149329828, 5.81425642275690049568553481297, 6.05511775830054759433995111157, 6.63361830396013314539378177155, 7.25047783802838427330028450541, 7.44557030498848387492737405058, 8.208681846377239693918583479320, 8.672365196683144779961098572501
