| L(s) = 1 | + 2·4-s + 5·7-s + 7·13-s + 7·19-s − 10·25-s + 10·28-s + 7·31-s + 37-s − 5·43-s + 18·49-s + 14·52-s − 14·61-s − 8·64-s − 11·67-s + 7·73-s + 14·76-s + 13·79-s + 35·91-s − 14·97-s − 20·100-s − 14·103-s − 17·109-s − 22·121-s + 14·124-s + 127-s + 131-s + 35·133-s + ⋯ |
| L(s) = 1 | + 4-s + 1.88·7-s + 1.94·13-s + 1.60·19-s − 2·25-s + 1.88·28-s + 1.25·31-s + 0.164·37-s − 0.762·43-s + 18/7·49-s + 1.94·52-s − 1.79·61-s − 64-s − 1.34·67-s + 0.819·73-s + 1.60·76-s + 1.46·79-s + 3.66·91-s − 1.42·97-s − 2·100-s − 1.37·103-s − 1.62·109-s − 2·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 3.03·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.411589761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.411589761\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.00618687603374488100522142787, −10.75743606721848065063443148425, −10.24505384462205628742717488056, −9.681627654740260582367234391236, −9.118792271025037367087670882804, −8.754741901892339896806207403149, −8.017022069291099811284592791469, −7.938533874516159532729217850229, −7.60965164553810521912058029929, −6.94145440879291088465465266815, −6.24926640933444087429668999049, −6.14595830660429113042814811759, −5.29940359658242321547915691426, −5.14099481787855597373490971154, −4.19359300362076988884737933878, −3.93017140693550642543459058897, −3.08517246852557650342965615941, −2.42952941470636248367231201601, −1.50383301377128624658897602765, −1.35894369580060168673200234568,
1.35894369580060168673200234568, 1.50383301377128624658897602765, 2.42952941470636248367231201601, 3.08517246852557650342965615941, 3.93017140693550642543459058897, 4.19359300362076988884737933878, 5.14099481787855597373490971154, 5.29940359658242321547915691426, 6.14595830660429113042814811759, 6.24926640933444087429668999049, 6.94145440879291088465465266815, 7.60965164553810521912058029929, 7.938533874516159532729217850229, 8.017022069291099811284592791469, 8.754741901892339896806207403149, 9.118792271025037367087670882804, 9.681627654740260582367234391236, 10.24505384462205628742717488056, 10.75743606721848065063443148425, 11.00618687603374488100522142787
