L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 2·9-s − 2·14-s + 16-s + 2·18-s + 4·19-s − 10·25-s + 2·28-s − 32-s − 2·36-s + 4·37-s − 4·38-s + 16·43-s + 3·49-s + 10·50-s + 12·53-s − 2·56-s − 4·63-s + 64-s + 2·72-s − 4·74-s + 4·76-s + 16·79-s − 5·81-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.534·14-s + 1/4·16-s + 0.471·18-s + 0.917·19-s − 2·25-s + 0.377·28-s − 0.176·32-s − 1/3·36-s + 0.657·37-s − 0.648·38-s + 2.43·43-s + 3/7·49-s + 1.41·50-s + 1.64·53-s − 0.267·56-s − 0.503·63-s + 1/8·64-s + 0.235·72-s − 0.464·74-s + 0.458·76-s + 1.80·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.183647893\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.183647893\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.139513650481559174344623349620, −8.527724601400848222788670647982, −8.305301817736640156017703661439, −7.57571100088867902110310233811, −7.52725309122624587441708234050, −6.87975810404488340043398375902, −6.10121749499441085588030360800, −5.57928681742950427486583645839, −5.51487510751237752431573865412, −4.42789561057506381074928638435, −4.05731801132236751489021266456, −3.20671223694547121384446442813, −2.49822445009788286784730652330, −1.86250960874307840989277735252, −0.795913931859384347135515195825,
0.795913931859384347135515195825, 1.86250960874307840989277735252, 2.49822445009788286784730652330, 3.20671223694547121384446442813, 4.05731801132236751489021266456, 4.42789561057506381074928638435, 5.51487510751237752431573865412, 5.57928681742950427486583645839, 6.10121749499441085588030360800, 6.87975810404488340043398375902, 7.52725309122624587441708234050, 7.57571100088867902110310233811, 8.305301817736640156017703661439, 8.527724601400848222788670647982, 9.139513650481559174344623349620