L(s) = 1 | − 2-s − 3·5-s − 7-s + 8-s + 3·10-s + 11-s + 4·13-s + 14-s − 16-s + 3·17-s − 2·19-s − 22-s − 3·23-s + 5·25-s − 4·26-s − 2·31-s − 3·34-s + 3·35-s − 8·37-s + 2·38-s − 3·40-s − 18·41-s − 8·43-s + 3·46-s − 3·47-s − 6·49-s − 5·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s + 0.301·11-s + 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.727·17-s − 0.458·19-s − 0.213·22-s − 0.625·23-s + 25-s − 0.784·26-s − 0.359·31-s − 0.514·34-s + 0.507·35-s − 1.31·37-s + 0.324·38-s − 0.474·40-s − 2.81·41-s − 1.21·43-s + 0.442·46-s − 0.437·47-s − 6/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2131375130\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2131375130\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−10.21405442887395465322949143242, −9.182614529256250193980075588850, −8.907002711422960175695821413280, −8.666728885530200608808099551456, −7.998059111966860772730307215758, −7.88124647680187511724442287363, −7.72155545162421651247777236655, −6.75950030300683092856086380864, −6.64411987099068353693731023810, −6.40892400549930428434063361189, −5.60471159350409321710190664479, −5.03707543088538741880961131365, −4.83674380366151270680791359249, −3.86754736033787239974227402154, −3.86057515678570666065059793922, −3.33054616202755432960690540686, −2.86772751291315523064126674682, −1.56799511130876613348899266478, −1.56796508133297508417498966473, −0.22007851049380365941551582820,
0.22007851049380365941551582820, 1.56796508133297508417498966473, 1.56799511130876613348899266478, 2.86772751291315523064126674682, 3.33054616202755432960690540686, 3.86057515678570666065059793922, 3.86754736033787239974227402154, 4.83674380366151270680791359249, 5.03707543088538741880961131365, 5.60471159350409321710190664479, 6.40892400549930428434063361189, 6.64411987099068353693731023810, 6.75950030300683092856086380864, 7.72155545162421651247777236655, 7.88124647680187511724442287363, 7.998059111966860772730307215758, 8.666728885530200608808099551456, 8.907002711422960175695821413280, 9.182614529256250193980075588850, 10.21405442887395465322949143242