L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 2·7-s − 5·9-s − 4·10-s + 2·11-s + 8·13-s + 4·14-s − 4·16-s + 10·18-s + 4·20-s − 4·22-s − 7·25-s − 16·26-s − 4·28-s + 14·31-s + 8·32-s − 4·35-s − 10·36-s − 12·43-s + 4·44-s − 10·45-s + 16·47-s − 3·49-s + 14·50-s + 16·52-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 0.755·7-s − 5/3·9-s − 1.26·10-s + 0.603·11-s + 2.21·13-s + 1.06·14-s − 16-s + 2.35·18-s + 0.894·20-s − 0.852·22-s − 7/5·25-s − 3.13·26-s − 0.755·28-s + 2.51·31-s + 1.41·32-s − 0.676·35-s − 5/3·36-s − 1.82·43-s + 0.603·44-s − 1.49·45-s + 2.33·47-s − 3/7·49-s + 1.97·50-s + 2.21·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8610581309\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8610581309\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.627153557857953172015491503977, −8.603539619290756001226038948684, −8.016470746894750837719454735098, −7.46973642361763444298598982747, −6.68253837392674763250909414891, −6.36261389471308870138602900888, −6.05856391223997850967309901807, −5.71143819833623906703502312707, −4.99767010522499384059415710151, −4.07790776750078242353026984533, −3.63865093449500368439278742924, −2.90247167943624077903906049478, −2.30747219091521399299623542576, −1.49733172832762767313391351767, −0.68419642869801904316651703930,
0.68419642869801904316651703930, 1.49733172832762767313391351767, 2.30747219091521399299623542576, 2.90247167943624077903906049478, 3.63865093449500368439278742924, 4.07790776750078242353026984533, 4.99767010522499384059415710151, 5.71143819833623906703502312707, 6.05856391223997850967309901807, 6.36261389471308870138602900888, 6.68253837392674763250909414891, 7.46973642361763444298598982747, 8.016470746894750837719454735098, 8.603539619290756001226038948684, 8.627153557857953172015491503977