L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 4·13-s + 5·16-s − 4·18-s + 8·19-s + 10·25-s − 8·26-s − 6·32-s + 6·36-s − 16·38-s − 16·43-s − 2·49-s − 20·50-s + 12·52-s + 12·53-s + 7·64-s + 16·67-s − 8·72-s + 24·76-s − 5·81-s + 32·86-s − 12·89-s + 4·98-s + 30·100-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2/3·9-s + 1.10·13-s + 5/4·16-s − 0.942·18-s + 1.83·19-s + 2·25-s − 1.56·26-s − 1.06·32-s + 36-s − 2.59·38-s − 2.43·43-s − 2/7·49-s − 2.82·50-s + 1.66·52-s + 1.64·53-s + 7/8·64-s + 1.95·67-s − 0.942·72-s + 2.75·76-s − 5/9·81-s + 3.45·86-s − 1.27·89-s + 0.404·98-s + 3·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.188881690\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.188881690\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−10.64896913563247714245311646407, −10.56793271005754251982042105805, −9.960876811290948746888204112180, −9.563932990430062457679054238899, −9.341239075667857016820254525143, −8.615096060217754910464155159115, −8.371326909021417023397777812427, −8.104149322960137769980037775363, −7.27094322719512629168055370326, −6.99155221989097526442870355040, −6.79379756635244739325087744273, −6.07441971692121125140075145350, −5.48806145679598763124225874359, −5.04441199958066535538761740686, −4.31721149977573346282347376739, −3.31409727277292343670850099827, −3.29287786882727944161359283658, −2.26481250631177369319880629320, −1.38660847017651924887315861907, −0.903121005584487045429289479848,
0.903121005584487045429289479848, 1.38660847017651924887315861907, 2.26481250631177369319880629320, 3.29287786882727944161359283658, 3.31409727277292343670850099827, 4.31721149977573346282347376739, 5.04441199958066535538761740686, 5.48806145679598763124225874359, 6.07441971692121125140075145350, 6.79379756635244739325087744273, 6.99155221989097526442870355040, 7.27094322719512629168055370326, 8.104149322960137769980037775363, 8.371326909021417023397777812427, 8.615096060217754910464155159115, 9.341239075667857016820254525143, 9.563932990430062457679054238899, 9.960876811290948746888204112180, 10.56793271005754251982042105805, 10.64896913563247714245311646407