L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 16-s + 2·19-s − 23-s + 3·31-s − 2·38-s + 40-s + 46-s − 2·47-s + 49-s − 2·53-s − 3·61-s − 3·62-s + 64-s + 3·79-s − 80-s + 3·83-s + 2·94-s + 2·95-s − 98-s + 2·106-s − 115-s + 121-s + 3·122-s + ⋯ |
L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 16-s + 2·19-s − 23-s + 3·31-s − 2·38-s + 40-s + 46-s − 2·47-s + 49-s − 2·53-s − 3·61-s − 3·62-s + 64-s + 3·79-s − 80-s + 3·83-s + 2·94-s + 2·95-s − 98-s + 2·106-s − 115-s + 121-s + 3·122-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9847161632\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9847161632\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.114380424932356413029279956265, −8.774639717621336695108187650658, −8.063613244514016110045635227022, −7.964293178622560157125849434103, −7.83330835762579070760500838464, −7.35701975827551266653767395344, −6.62366266728923752217915642105, −6.52888064506062083826810214654, −6.09157546690649956605185555807, −5.74538172411086929913050377280, −5.07868091854618054292975873012, −4.84980638737296705692982102553, −4.60308608784609179346475750161, −3.95428718467989718258389281024, −3.24176508697820678014290675531, −3.12133694147074461780703827854, −2.33271783431510951357366550408, −1.87788608412848438253145007340, −1.35224497254611265376290807971, −0.78878678362419557792982122383,
0.78878678362419557792982122383, 1.35224497254611265376290807971, 1.87788608412848438253145007340, 2.33271783431510951357366550408, 3.12133694147074461780703827854, 3.24176508697820678014290675531, 3.95428718467989718258389281024, 4.60308608784609179346475750161, 4.84980638737296705692982102553, 5.07868091854618054292975873012, 5.74538172411086929913050377280, 6.09157546690649956605185555807, 6.52888064506062083826810214654, 6.62366266728923752217915642105, 7.35701975827551266653767395344, 7.83330835762579070760500838464, 7.964293178622560157125849434103, 8.063613244514016110045635227022, 8.774639717621336695108187650658, 9.114380424932356413029279956265